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## How to Solve Any Physics Problem

Last Updated: July 21, 2023 Fact Checked

This article was co-authored by Sean Alexander, MS . Sean Alexander is an Academic Tutor specializing in teaching mathematics and physics. Sean is the Owner of Alexander Tutoring, an academic tutoring business that provides personalized studying sessions focused on mathematics and physics. With over 15 years of experience, Sean has worked as a physics and math instructor and tutor for Stanford University, San Francisco State University, and Stanbridge Academy. He holds a BS in Physics from the University of California, Santa Barbara and an MS in Theoretical Physics from San Francisco State University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 328,635 times.

Baffled as to where to begin with a physics problem? There is a very simply and logical flow process to solving any physics problem.

• Ask yourself if your answers make sense. If the numbers look absurd (for example, you get that a rock dropped off a 50-meter cliff moves with the speed of only 0.00965 meters per second when it hits the ground), you made a mistake somewhere.
• Don't forget to include the units into your answers, and always keep track of them. So, if you are solving for velocity and get your answer in seconds, that is a sign that something went wrong, because it should be in meters per second.
• Plug your answers back into the original equations to make sure you get the same number on both sides.

## Community Q&A

• Many people report that if they leave a problem for a while and come back to it later, they find they have a new perspective on it and can sometimes see an easy way to the answer that they did not notice before. Thanks Helpful 249 Not Helpful 48
• Try to understand the problem first. Thanks Helpful 186 Not Helpful 51
• Remember, the physics part of the problem is figuring out what you are solving for, drawing the diagram, and remembering the formulae. The rest is just use of algebra, trigonometry, and/or calculus, depending on the difficulty of your course. Thanks Helpful 115 Not Helpful 34

• Physics is not easy to grasp for many people, so do not get bent out of shape over a problem. Thanks Helpful 100 Not Helpful 25
• If an instructor tells you to draw a free body diagram, be sure that that is exactly what you draw. Thanks Helpful 89 Not Helpful 24

## Things You'll Need

• A Writing Utensil (preferably a pencil or erasable pen of sorts)
• Calculator with all the functions you need for your exam
• An understanding of the equations needed to solve the problems. Or a list of them will suffice if you are just trying to get through the course alive.

## Expert Interview

Thanks for reading our article! If you’d like to learn more about teaching, check out our in-depth interview with Sean Alexander, MS .

• ↑ https://iopscience.iop.org/article/10.1088/1361-6404/aa9038
• ↑ https://physics.wvu.edu/files/d/ce78505d-1426-4d68-8bb2-128d8aac6b1b/expertapproachtosolvingphysicsproblems.pdf
• ↑ https://www.brighthubeducation.com/science-homework-help/42596-tips-to-choosing-the-correct-physics-formula/

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## Example Physics Problems and Solutions

Learning how to solve physics problems is a big part of learning physics. Here’s a collection of example physics problems and solutions to help you tackle problems sets and understand concepts and how to work with formulas:

Physics Homework Tips Physics homework can be challenging! Get tips to help make the task a little easier.

## Unit Conversion Examples

There are now too many unit conversion examples to list in this space. This Unit Conversion Examples page is a more comprehensive list of worked example problems.

## Newton’s Equations of Motion Example Problems

Equations of Motion – Constant Acceleration Example This equations of motion example problem consist of a sliding block under constant acceleration. It uses the equations of motion to calculate the position and velocity of a given time and the time and position of a given velocity.

Equations of Motion Example Problem – Constant Acceleration This example problem uses the equations of motion for constant acceleration to find the position, velocity, and acceleration of a breaking vehicle.

Equations of Motion Example Problem – Interception

This example problem uses the equations of motion for constant acceleration to calculate the time needed for one vehicle to intercept another vehicle moving at a constant velocity.

Vertical Motion Example Problem – Coin Toss Here’s an example applying the equations of motion under constant acceleration to determine the maximum height, velocity and time of flight for a coin flipped into a well. This problem could be modified to solve any object tossed vertically or dropped off a tall building or any height. This type of problem is a common equation of motion homework problem.

Projectile Motion Example Problem This example problem shows how to find different variables associated with parabolic projectile motion.

Accelerometer and Inertia Example Problem Accelerometers are devices to measure or detect acceleration by measuring the changes that occur as a system experiences an acceleration. This example problem uses one of the simplest forms of an accelerometer, a weight hanging from a stiff rod or wire. As the system accelerates, the hanging weight is deflected from its rest position. This example derives the relationship between that angle, the acceleration and the acceleration due to gravity. It then calculates the acceleration due to gravity of an unknown planet.

Weight In An Elevator Have you ever wondered why you feel slightly heavier in an elevator when it begins to move up? Or why you feel lighter when the elevator begins to move down? This example problem explains how to find your weight in an accelerating elevator and how to find the acceleration of an elevator using your weight on a scale.

Equilibrium Example Problem This example problem shows how to determine the different forces in a system at equilibrium. The system is a block suspended from a rope attached to two other ropes.

Equilibrium Example Problem – Balance This example problem highlights the basics of finding the forces acting on a system in mechanical equilibrium.

Force of Gravity Example This physics problem and solution shows how to apply Newton’s equation to calculate the gravitational force between the Earth and the Moon.

## Coupled Systems Example Problems

Coupled systems are two or more separate systems connected together. The best way to solve these types of problems is to treat each system separately and then find common variables between them. Atwood Machine The Atwood Machine is a coupled system of two weights sharing a connecting string over a pulley. This example problem shows how to find the acceleration of an Atwood system and the tension in the connecting string. Coupled Blocks – Inertia Example This example problem is similar to the Atwood machine except one block is resting on a frictionless surface perpendicular to the other block. This block is hanging over the edge and pulling down on the coupled string. The problem shows how to calculate the acceleration of the blocks and the tension in the connecting string.

## Friction Example Problems

These example physics problems explain how to calculate the different coefficients of friction.

Friction Example Problem – Block Resting on a Surface Friction Example Problem – Coefficient of Static Friction Friction Example Problem – Coefficient of Kinetic Friction Friction and Inertia Example Problem

## Momentum and Collisions Example Problems

These example problems show how to calculate the momentum of moving masses.

Momentum and Impulse Example Finds the momentum before and after a force acts on a body and determine the impulse of the force.

Elastic Collision Example Shows how to find the velocities of two masses after an elastic collision.

It Can Be Shown – Elastic Collision Math Steps Shows the math to find the equations expressing the final velocities of two masses in terms of their initial velocities.

## Simple Pendulum Example Problems

These example problems show how to use the period of a pendulum to find related information.

Find the Period of a Simple Pendulum Find the period if you know the length of a pendulum and the acceleration due to gravity.

Find the Length of a Simple Pendulum Find the length of the pendulum when the period and acceleration due to gravity is known.

Find the Acceleration due to Gravity Using A Pendulum Find ‘g’ on different planets by timing the period of a known pendulum length.

## Harmonic Motion and Waves Example Problems

These example problems all involve simple harmonic motion and wave mechanics.

Energy and Wavelength Example This example shows how to determine the energy of a photon of a known wavelength.

Hooke’s Law Example Problem An example problem involving the restoring force of a spring.

Wavelength and Frequency Calculations See how to calculate wavelength if you know frequency and vice versa, for light, sound, or other waves.

## Heat and Energy Example Problems

Heat of Fusion Example Problem Two example problems using the heat of fusion to calculate the energy required for a phase change.

Specific Heat Example Problem This is actually 3 similar example problems using the specific heat equation to calculate heat, specific heat, and temperature of a system.

Heat of Vaporization Example Problems Two example problems using or finding the heat of vaporization.

Ice to Steam Example Problem Classic problem melting cold ice to make hot steam. This problem brings all three of the previous example problems into one problem to calculate heat changes over phase changes.

## Charge and Coulomb Force Example Problems

Electrical charges generate a coulomb force between themselves proportional to the magnitude of the charges and inversely proportional to the distance between them. Coulomb’s Law Example This example problem shows how to use Coulomb’s Law equation to find the charges necessary to produce a known repulsive force over a set distance. Coulomb Force Example This Coulomb force example shows how to find the number of electrons transferred between two bodies to generate a set amount of force over a short distance.

## 6.1 Solving Problems with Newton’s Laws

Learning objectives.

By the end of this section, you will be able to:

• Apply problem-solving techniques to solve for quantities in more complex systems of forces
• Use concepts from kinematics to solve problems using Newton’s laws of motion
• Solve more complex equilibrium problems
• Solve more complex acceleration problems
• Apply calculus to more advanced dynamics problems

Success in problem solving is necessary to understand and apply physical principles. We developed a pattern of analyzing and setting up the solutions to problems involving Newton’s laws in Newton’s Laws of Motion ; in this chapter, we continue to discuss these strategies and apply a step-by-step process.

## Problem-Solving Strategies

We follow here the basics of problem solving presented earlier in this text, but we emphasize specific strategies that are useful in applying Newton’s laws of motion . Once you identify the physical principles involved in the problem and determine that they include Newton’s laws of motion, you can apply these steps to find a solution. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, so the following techniques should reinforce skills you have already begun to develop.

## Problem-Solving Strategy

Applying newton’s laws of motion.

• Identify the physical principles involved by listing the givens and the quantities to be calculated.
• Sketch the situation, using arrows to represent all forces.
• Determine the system of interest. The result is a free-body diagram that is essential to solving the problem.
• Apply Newton’s second law to solve the problem. If necessary, apply appropriate kinematic equations from the chapter on motion along a straight line.
• Check the solution to see whether it is reasonable.

Let’s apply this problem-solving strategy to the challenge of lifting a grand piano into a second-story apartment. Once we have determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation. Such a sketch is shown in Figure 6.2 (a). Then, as in Figure 6.2 (b), we can represent all forces with arrows. Whenever sufficient information exists, it is best to label these arrows carefully and make the length and direction of each correspond to the represented force.

As with most problems, we next need to identify what needs to be determined and what is known or can be inferred from the problem as stated, that is, make a list of knowns and unknowns. It is particularly crucial to identify the system of interest, since Newton’s second law involves only external forces. We can then determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 6.2 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated in Newton’s Laws of Motion , the system of interest depends on the question we need to answer. Only forces are shown in free-body diagrams, not acceleration or velocity. We have drawn several free-body diagrams in previous worked examples. Figure 6.2 (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.

Once a free-body diagram is drawn, we apply Newton’s second law. This is done in Figure 6.2 (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then the forces can be handled algebraically. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. We do this by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known. Generally, just write Newton’s second law in components along the different directions. Then, you have the following equations:

(If, for example, the system is accelerating horizontally, then you can then set a y = 0 . a y = 0 . ) We need this information to determine unknown forces acting on a system.

As always, we must check the solution. In some cases, it is easy to tell whether the solution is reasonable. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving; with experience, it becomes progressively easier to judge whether an answer is reasonable. Another way to check a solution is to check the units. If we are solving for force and end up with units of millimeters per second, then we have made a mistake.

There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills. We look first at problems involving particle equilibrium, which make use of Newton’s first law, and then consider particle acceleration, which involves Newton’s second law.

## Particle Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration, but it is important to remember that these conditions are relative. For example, an object may be at rest when viewed from our frame of reference, but the same object would appear to be in motion when viewed by someone moving at a constant velocity. We now make use of the knowledge attained in Newton’s Laws of Motion , regarding the different types of forces and the use of free-body diagrams, to solve additional problems in particle equilibrium .

## Example 6.1

Different tensions at different angles.

Thus, as you might expect,

This gives us the following relationship:

Note that T 1 T 1 and T 2 T 2 are not equal in this case because the angles on either side are not equal. It is reasonable that T 2 T 2 ends up being greater than T 1 T 1 because it is exerted more vertically than T 1 . T 1 .

Now consider the force components along the vertical or y -axis:

This implies

Substituting the expressions for the vertical components gives

There are two unknowns in this equation, but substituting the expression for T 2 T 2 in terms of T 1 T 1 reduces this to one equation with one unknown:

which yields

Solving this last equation gives the magnitude of T 1 T 1 to be

Finally, we find the magnitude of T 2 T 2 by using the relationship between them, T 2 = 1.225 T 1 T 2 = 1.225 T 1 , found above. Thus we obtain

## Significance

Particle acceleration.

We have given a variety of examples of particles in equilibrium. We now turn our attention to particle acceleration problems, which are the result of a nonzero net force. Refer again to the steps given at the beginning of this section, and notice how they are applied to the following examples.

## Example 6.2

Drag force on a barge.

The drag of the water F → D F → D is in the direction opposite to the direction of motion of the boat; this force thus works against F → app , F → app , as shown in the free-body diagram in Figure 6.4 (b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Because the applied forces are perpendicular, the x - and y -axes are in the same direction as F → 1 F → 1 and F → 2 . F → 2 . The problem quickly becomes a one-dimensional problem along the direction of F → app F → app , since friction is in the direction opposite to F → app . F → app . Our strategy is to find the magnitude and direction of the net applied force F → app F → app and then apply Newton’s second law to solve for the drag force F → D . F → D .

The angle is given by

From Newton’s first law, we know this is the same direction as the acceleration. We also know that F → D F → D is in the opposite direction of F → app , F → app , since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as F → app , F → app , but its magnitude is slightly less than F → app . F → app . The problem is now one-dimensional. From the free-body diagram, we can see that

However, Newton’s second law states that

This can be solved for the magnitude of the drag force of the water F D F D in terms of known quantities:

Substituting known values gives

The direction of F → D F → D has already been determined to be in the direction opposite to F → app , F → app , or at an angle of 53 ° 53 ° south of west.

In Newton’s Laws of Motion , we discussed the normal force , which is a contact force that acts normal to the surface so that an object does not have an acceleration perpendicular to the surface. The bathroom scale is an excellent example of a normal force acting on a body. It provides a quantitative reading of how much it must push upward to support the weight of an object. But can you predict what you would see on the dial of a bathroom scale if you stood on it during an elevator ride? Will you see a value greater than your weight when the elevator starts up? What about when the elevator moves upward at a constant speed? Take a guess before reading the next example.

## Example 6.3

What does the bathroom scale read in an elevator.

From the free-body diagram, we see that F → net = F → s − w → , F → net = F → s − w → , so we have

Solving for F s F s gives us an equation with only one unknown:

or, because w = m g , w = m g , simply

No assumptions were made about the acceleration, so this solution should be valid for a variety of accelerations in addition to those in this situation. ( Note: We are considering the case when the elevator is accelerating upward. If the elevator is accelerating downward, Newton’s second law becomes F s − w = − m a . F s − w = − m a . )

• We have a = 1.20 m/s 2 , a = 1.20 m/s 2 , so that F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) yielding F s = 825 N . F s = 825 N .
• Now, what happens when the elevator reaches a constant upward velocity? Will the scale still read more than his weight? For any constant velocity—up, down, or stationary—acceleration is zero because a = Δ v Δ t a = Δ v Δ t and Δ v = 0 . Δ v = 0 . Thus, F s = m a + m g = 0 + m g F s = m a + m g = 0 + m g or F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , which gives F s = 735 N . F s = 735 N .

Thus, the scale reading in the elevator is greater than his 735-N (165-lb.) weight. This means that the scale is pushing up on the person with a force greater than his weight, as it must in order to accelerate him upward. Clearly, the greater the acceleration of the elevator, the greater the scale reading, consistent with what you feel in rapidly accelerating versus slowly accelerating elevators. In Figure 6.5 (b), the scale reading is 735 N, which equals the person’s weight. This is the case whenever the elevator has a constant velocity—moving up, moving down, or stationary.

Now calculate the scale reading when the elevator accelerates downward at a rate of 1.20 m/s 2 . 1.20 m/s 2 .

The solution to the previous example also applies to an elevator accelerating downward, as mentioned. When an elevator accelerates downward, a is negative, and the scale reading is less than the weight of the person. If a constant downward velocity is reached, the scale reading again becomes equal to the person’s weight. If the elevator is in free fall and accelerating downward at g , then the scale reading is zero and the person appears to be weightless.

## Example 6.4

Two attached blocks.

For block 1: T → + w → 1 + N → = m 1 a → 1 T → + w → 1 + N → = m 1 a → 1

For block 2: T → + w → 2 = m 2 a → 2 . T → + w → 2 = m 2 a → 2 .

Notice that T → T → is the same for both blocks. Since the string and the pulley have negligible mass, and since there is no friction in the pulley, the tension is the same throughout the string. We can now write component equations for each block. All forces are either horizontal or vertical, so we can use the same horizontal/vertical coordinate system for both objects

When block 1 moves to the right, block 2 travels an equal distance downward; thus, a 1 x = − a 2 y . a 1 x = − a 2 y . Writing the common acceleration of the blocks as a = a 1 x = − a 2 y , a = a 1 x = − a 2 y , we now have

From these two equations, we can express a and T in terms of the masses m 1 and m 2 , and g : m 1 and m 2 , and g :

Calculate the acceleration of the system, and the tension in the string, when the masses are m 1 = 5.00 kg m 1 = 5.00 kg and m 2 = 3.00 kg . m 2 = 3.00 kg .

## Example 6.5

Atwood machine.

• We have For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . (The negative sign in front of m 2 a m 2 a indicates that m 2 m 2 accelerates downward; both blocks accelerate at the same rate, but in opposite directions.) Solve the two equations simultaneously (subtract them) and the result is ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . Solving for a : a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 . a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 .
• Observing the first block, we see that T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N . T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N .

Determine a general formula in terms of m 1 , m 2 m 1 , m 2 and g for calculating the tension in the string for the Atwood machine shown above.

## Newton’s Laws of Motion and Kinematics

Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics , and hence the relevance of earlier chapters.

When approaching problems that involve various types of forces, acceleration, velocity, and/or position, listing the givens and the quantities to be calculated will allow you to identify the principles involved. Then, you can refer to the chapters that deal with a particular topic and solve the problem using strategies outlined in the text. The following worked example illustrates how the problem-solving strategy given earlier in this chapter, as well as strategies presented in other chapters, is applied to an integrated concept problem.

## Example 6.6

What force must a soccer player exert to reach top speed.

• We are given the initial and final velocities (zero and 8.00 m/s forward); thus, the change in velocity is Δ v = 8.00 m/s Δ v = 8.00 m/s . We are given the elapsed time, so Δ t = 2.50 s . Δ t = 2.50 s . The unknown is acceleration, which can be found from its definition: a = Δ v Δ t . a = Δ v Δ t . Substituting the known values yields a = 8.00 m/s 2.50 s = 3.20 m/s 2 . a = 8.00 m/s 2.50 s = 3.20 m/s 2 .
• Here we are asked to find the average force the ground exerts on the runner to produce this acceleration. (Remember that we are dealing with the force or forces acting on the object of interest.) This is the reaction force to that exerted by the player backward against the ground, by Newton’s third law. Neglecting air resistance, this would be equal in magnitude to the net external force on the player, since this force causes her acceleration. Since we now know the player’s acceleration and are given her mass, we can use Newton’s second law to find the force exerted. That is, F net = m a . F net = m a . Substituting the known values of m and a gives F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N . F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N .

This is a reasonable result: The acceleration is attainable for an athlete in good condition. The force is about 50 pounds, a reasonable average force.

The soccer player stops after completing the play described above, but now notices that the ball is in position to be stolen. If she now experiences a force of 126 N to attempt to steal the ball, which is 2.00 m away from her, how long will it take her to get to the ball?

## Example 6.7

What force acts on a model helicopter.

The magnitude of the force is now easily found:

Find the direction of the resultant for the 1.50-kg model helicopter.

## Example 6.8

Baggage tractor.

• ∑ F x = m system a x ∑ F x = m system a x and ∑ F x = 820.0 t , ∑ F x = 820.0 t , so 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . Since acceleration is a function of time, we can determine the velocity of the tractor by using a = d v d t a = d v d t with the initial condition that v 0 = 0 v 0 = 0 at t = 0 . t = 0 . We integrate from t = 0 t = 0 to t = 3 : t = 3 : d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s . d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s .
• Refer to the free-body diagram in Figure 6.8 (b). ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N . ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N .

Recall that v = d s d t v = d s d t and a = d v d t a = d v d t . If acceleration is a function of time, we can use the calculus forms developed in Motion Along a Straight Line , as shown in this example. However, sometimes acceleration is a function of displacement. In this case, we can derive an important result from these calculus relations. Solving for dt in each, we have d t = d s v d t = d s v and d t = d v a . d t = d v a . Now, equating these expressions, we have d s v = d v a . d s v = d v a . We can rearrange this to obtain a d s = v d v . a d s = v d v .

## Example 6.9

Motion of a projectile fired vertically.

The acceleration depends on v and is therefore variable. Since a = f ( v ) , a = f ( v ) , we can relate a to v using the rearrangement described above,

We replace ds with dy because we are dealing with the vertical direction,

We now separate the variables ( v ’s and dv ’s on one side; dy on the other):

Thus, h = 114 m . h = 114 m .

If atmospheric resistance is neglected, find the maximum height for the mortar shell. Is calculus required for this solution?

## Interactive

Explore the forces at work in this simulation when you try to push a filing cabinet. Create an applied force and see the resulting frictional force and total force acting on the cabinet. Charts show the forces, position, velocity, and acceleration vs. time. View a free-body diagram of all the forces (including gravitational and normal forces).

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• Book title: University Physics Volume 1
• Publication date: Sep 19, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• Section URL: https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws

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## Dimensional Analysis

Any physical quantity can be expressed as a product of a combination of the basic physical dimensions.

learning objectives

• Calculate the conversion from one kind of dimension to another

The dimension of a physical quantity indicates how it relates to one of the seven basic quantities. These fundamental quantities are:

• [A] Current
• [K] Temperature
• [mol] Amount of a Substance
• [cd] Luminous Intensity

As you can see, the symbol is enclosed in a pair of square brackets. This is often used to represent the dimension of individual basic quantity. An example of the use of basic dimensions is speed, which has a dimension of 1 in length and -1 in time; $$\mathrm{\frac{[L]}{[T]}=[LT^{−1}]}$$. Any physical quantity can be expressed as a product of a combination of the basic physical dimensions.

Dimensional analysis is the practice of checking relations between physical quantities by identifying their dimensions. The dimension of any physical quantity is the combination of the basic physical dimensions that compose it. Dimensional analysis is based on the fact that physical law must be independent of the units used to measure the physical variables. It can be used to check the plausibility of derived equations, computations and hypotheses.

## Derived Dimensions

The dimensions of derived quantities may include few or all dimensions in individual basic quantities. In order to understand the technique to write dimensions of a derived quantity, we consider the case of force. Force is defined as:

\begin{align} \mathrm{F} &= \mathrm{m⋅a} \\ \mathrm{F} &= \mathrm{[M][a]} \end{align}

The dimension of acceleration, represented as [a], is itself a derived quantity being the ratio of velocity and time. In turn, velocity is also a derived quantity, being ratio of length and time.

\begin{align} \mathrm{F} &= \mathrm{[M][a]=[M][vT^{−1}]} \\ \mathrm{F} &= \mathrm{[M][LT^{−1}T^{−1}]=[MLT^{−2}]} \end{align}

## Dimensional Conversion

In practice, one might need to convert from one kind of dimension to another. For common conversions, you might already know how to convert off the top of your head. But for less common ones, it is helpful to know how to find the conversion factor:

$\mathrm{Q=n_1u_1=n_2u_2}$

where n represents the amount per u dimensions. You can then use ratios to figure out the conversion:

$\mathrm{n_2=\dfrac{u_2}{u_1}⋅n_1}$

## Trigonometry

Trigonometry is central to the use of free body diagrams, which help visually represent difficult physics problems.

• Explain why trigonometry is useful in determining horizontal and vertical components of forces

## Trigonometry and Solving Physics Problems

In physics, most problems are solved much more easily when a free body diagram is used. Free body diagrams use geometry and vectors to visually represent the problem. Trigonometry is also used in determining the horizontal and vertical components of forces and objects. Free body diagrams are very helpful in visually identifying which components are unknown and where the moments are applied. They can help analyze a problem, whether it is static or dynamic.

When people draw free body diagrams, often not everything is perfectly parallel and perpendicular. Sometimes people need to analyze the horizontal and vertical components of forces and object orientation. When the force or object is not acting parallel to the x or y axis, people can employ basic trigonometry to use the simplest components of the action to analyze it. Basically, everything should be considered in terms of x and y , which sometimes takes some manipulation.

Free Body Diagram : The rod is hinged from a wall and is held with the help of a string.

A rod ‘AB’ is hinged at ‘A’ from a wall and is held still with the help of a string, as shown in. This exercise involves drawing the free body diagram. To make the problem easier, the force F will be expressed in terms of its horizontal and vertical components. Removing all other elements from the image helps produce the finished free body diagram.

Free Body Diagram : The free body diagram as a finished product

Given the finished free body diagram, people can use their knowledge of trigonometry and the laws of sine and cosine to mathematically and numerical represent the horizontal and vertical components:

## General Problem-Solving Tricks

Free body diagrams use geometry and vectors to visually represent the problem.

• Construct a free-body diagram for a physical scenario

In physics, most problems are solved much more easily when a free body diagram is used. This uses geometry and vectors to visually represent to problem, and trigonometry is also used in determining horizontal and vertical components of forces and objects.

Purpose: Free body diagrams are very helpful in visually identifying which components are unknown, where the moments are applied, and help analyze a problem, whether static or dynamic.

## How to Make A Free Body Diagram

To draw a free body diagram, do not worry about drawing it to scale, this will just be what you use to help yourself identify the problems. First you want to model the body, in one of three ways:

• As a particle. This model may be used when any turning effects are zero or have zero interest even though the body itself may be extended. The body may be represented by a small symbolic blob and the diagram reduces to a set of concurrent arrows. A force on a particle is a bound vector.
• rigid extended . Stresses and strains are of no interest but turning effects are. A force arrow should lie along the line of force, but where along the line is irrelevant. A force on an extended rigid body is a sliding vector.
• non-rigid extended . The point of application of a force becomes crucial and has to be indicated on the diagram. A force on a non-rigid body is a bound vector. Some engineers use the tail of the arrow to indicate the point of application. Others use the tip.

## Do’s and Don’ts

What to include: Since a free body diagram represents the body itself and the external forces on it. So you will want to include the following things in the diagram:

• The body: This is usually sketched in a schematic way depending on the body – particle/extended, rigid/non-rigid – and on what questions are to be answered. Thus if rotation of the body and torque is in consideration, an indication of size and shape of the body is needed.
• The external forces: These are indicated by labelled arrows. In a fully solved problem, a force arrow is capable of indicating the direction, the magnitude the point of application. These forces can be friction, gravity, normal force, drag, tension, etc…

## Do not include:

• Do not show bodies other than the body of interest.
• Do not show forces exerted by the body.
• Internal forces acting on various parts of the body by other parts of the body.
• Any velocity or acceleration is left out.

How To Solve Any Physics Problem : Learn five simple steps in five minutes! In this episode we cover the most effective problem-solving method I’ve encountered and call upon some fuzzy friends to help us remember the steps.

Free Body Diagram : Use this figure to work through the example problem.

• Dimensional analysis is the practice of checking relations amount physical quantities by identifying their dimensions.
• It is common to be faced with a problem that uses different dimensions to express the same basic quantity. The following equation can be used to find the conversion factor between the two derived dimensions: $$\mathrm{n_2=\frac{u_2}{u_1} \times n_1}$$.
• Dimensional analysis can also be used as a simple check to computations, theories and hypotheses.
• It is important to identify the problem and the unknowns and draw them in a free body diagram.
• The laws of cosine and sine can be used to determine the vertical and horizontal components of the different elements of the diagram.
• Free body diagrams use geometry and vectors to visually represent physics problems.
• A free body diagram lets you visually isolate the problem you are trying to solve, and simplify it into simple geometry and trigonometry.
• When drawing these diagrams, it is helpful to only draw the body it self, and the forces acting on it.
• Drawing other objects and internal forces can condense the diagram and cause it to be less helpful.
• dimension : A measure of spatial extent in a particular direction, such as height, width or breadth, or depth.
• trigonometry : The branch of mathematics that deals with the relationships between the sides and the angles of triangles and the calculations based on them, particularly the trigonometric functions.
• static : Fixed in place; having no motion.
• dynamic : Changing; active; in motion.

• Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike

• Dimensional analysis. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Dimensional_analysis . License : CC BY-SA: Attribution-ShareAlike
• Sunil Kumar Singh, Dimensional Analysis. September 18, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m15037/latest/ . License : CC BY: Attribution
• dimension. Provided by : Wiktionary. Located at : http://en.wiktionary.org/wiki/dimension . License : CC BY-SA: Attribution-ShareAlike
• Sunil Kumar Singh, Free Body Diagram (Application). September 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m14720/latest/ . License : CC BY: Attribution
• trigonometry. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/trigonometry . License : CC BY-SA: Attribution-ShareAlike
• Sunil Kumar Singh, Free Body Diagram (Application). February 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m14720/latest/ . License : CC BY: Attribution
• Free body diagram. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Free_body_diagram . License : CC BY-SA: Attribution-ShareAlike
• dynamic. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/dynamic . License : CC BY-SA: Attribution-ShareAlike
• static. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/static . License : CC BY-SA: Attribution-ShareAlike
• Free Body Diagram. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Free_Body_Diagram.png . License : CC BY-SA: Attribution-ShareAlike

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## 4 tricks for solving any physics problem

Physics can be intimidating—all those pulleys and protons and projectile motion. If you approach it with the right mindset, however, even the hardest problems are usually easier than you think. When you come up against a tough question, don’t panic. Instead, start with these short, easy tricks to help you work through the problem.

## 4 tricks for solving any physics problem:

1. what is the subject.

Just about every physics question is testing specific knowledge. When you read the question ask yourself, is it exploring electricity? Torque? Parabolic motion? Each topic is associated with specific equations and approaches, so recognizing the subject will focus your effort in the right direction. Look for keywords and phrases that reveal the topic.

## 2. What are you trying to find?

This simple step can save a lot of time. Before starting to solve the problem, think about what the answer will look like. What are the units; is the final answer going to be in kilograms or liters? Also, consider what other physical quantities might relate to your answer. If you’re trying to find speed, it might be useful to find acceleration, then solve that for speed. Determining restrictions on the answer early also ensures you answer the specific question; a common mistake in physics is solving for the wrong thing.

## 3. What do you know?

Think about what details the problem mentions. Unless the question is really bad, they probably gave you exactly the information you need to solve the problem. Don’t be surprised if sometimes this information is coded in language; a problem that mentions a spring with “the mass removed from the end” is telling you something important about the quantities of force. Write down every quantity you know from the problem, then proceed to…

## 4. What equations can you use?

What equations include the quantities you know and also the one you’re looking for? If you have the mass of an object and a force and you’re trying to find the acceleration, start with F=ma (Newton’s second law). If you’re trying to find the electric field but you have the charge and the distance, try E=q/(4πε*r 2 ).

If you’re having trouble figuring out which equation to use, go back to our first trick. What equations are associated with the topic? Can you manipulate the quantities you have to fit in any of them?

## Bonus Trick: “hack” the units

This trick doesn’t always work but it can jumpstart your brain. First, determine the units of the quantity you’re trying to find and the quantities you have. Only use base units (meters, kilograms, seconds, charge), not compound units (Force is measured in Newtons, which are just kg*m/s 2 ). Multiply and divide the quantities until the units match the units of the answer quantity. For example, if you’re trying to find Potential Energy (kg*m 2 /s 2 ) and you have the height (m), mass (kg), and gravitational acceleration (m/s 2 ), you can match the units by multiplying the three quantities (m*kg*m/s 2 =kg*m 2 /s 2 ).

Note: Unlike the other ones, this trick won’t always work. Watch out for unitless constants. For example, Kinetic energy is ½*mass*velocity 2 , not just mass*velocity 2 as the units suggest. Even though this trick isn’t perfect, however, it can still be a great place to start.

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•    How to Study Physics: 15 Killer Strategies to Boost Your Grades

## Apply as a tutor to teach students online from anywhere in the world.

• Chloe Daniel
• Published On: March 12 ,2024

When it comes to physics, students can’t wait to get rid of the subject because it is packed with so much math and theories. But the truth is that physics is as interesting as other science subjects. The physics field has allowed many industries to succeed and has played a huge part in the world we live in today. If you think about it, no technology is possible without physics.

Now, let’s come back to the point. Today’s blog will uncover 15 super simple ways to be better at physics. After you’re done reading the topic, you will feel much more confident.

## 15 Tips to Learn Physics Better

1. ace the basics.

As you already know, this science subject is based on theories through which all other concepts are connected. This means that whatever physics problem you will solve in a GCSE exam will use the key principles and basic concepts of the subject.

The biggest mistake a physics student can make is memorizing difficult problems. Doing this only makes them assume that physics is the hardest subject ever. Therefore, it is advised to master the basics first in order to know how all concepts are connected.

So, what’s the best way to know the basics? Make a mind map! This trick links a subject’s concepts. What’s more, you need to retain the core physics equations. Here are some examples:

• Average speed: Total distance / Total time
• Density: Mass of the body / volume of the body
• Work = Displacement x force
• Force = Mass x acceleration
• Momentum = Mass x velocity

You should also know how these equations work and are applied. This is how students can easily solve complicated physics problems.

## 2. Know How Key Equations Were Made

Once you have memorized the basic physics equations, the next step is knowing how and why they work. Basically, students should know how core equations are derived.

Knowing this will show the relationship between the equations in any grade. After learning how to derive the equations, you will know how to use them in solving problems.

## 3. Improve Your Math Skills

Physics has a close relation to math and numbers. This means that if you are poor in math, you can’t expect your physics grades to be any different. So, start with math first if you want to be praised by your physics teacher.

You can also study physics and math side-by-side. Either hire a credible tutor or join a study group. This will help you in knowing what and how to learn physics. Remember to give attention to the following math topics:

• Geometry for physics problems related to volume, area, and more.
• Algebra for basic physics equations.
• Trigonometry for angled systems, rotation problems, and force diagrams.

## 4. Always Keep an Eye For Small Details

Every physics problem is an example of a real-world situation, and these simplify how things operate to understand the situation. But this also means that the forces that influence the problem’s answer will be intentionally removed, like volume and friction.

In the majority of cases, the tutor will ignore the details in order to test your knowledge. This further proves the fact that students need to memorize minor details to guarantee a correct answer.

## 5. Simplify the Situations

In physics, it is not rare to understand a problem because of how difficult it is. But regardless of the problem, try to simplify it as much as possible. This can happen when you have another look at the problem and analyze it.

Soon enough, it will seem easier. Bring the problem to a familiar situation by simplifying it. If you find it hard to remain calm, dissect the problem by writing it down and making sections out of it. Solve each section and take your time.

Sometimes, physics problems also add details that are not needed for solutions. So, if you want to be better at solving physics problems, learn how to recognize important details only.

Tip: Write the relevant constants and equations and assign every part of the necessary details to the suitable variables.

At the end of every problem, give yourself time to double-check the solution. On average, leave half an hour in exam time to go through all your answers. Students in higher grades are aware that almost all physics problems include multiple mathematical calculations. If you make a mistake in even one calculation, you can’t get a correct answer. So, if you don’t want to lose the chances of a good grade, always double-check your answers.

Luckily, students can also use common sense to guess an answer. For instance, if you have to find the momentum of a forward-moving object, the answer can’t be negative.

## 7. Be Attentive in Class

Regardless of how your school teacher or tutor gives lectures, physics is a dull subject. It is so common for students to be distracted. But remember that when your mind wanders off, you will skip critical information.

Read these tips to have a longer attention span in physics classes:

• Read before your physics class to know the overall idea of the topic.
• Take important notes during lectures.
• Ask your teacher if the lesson can be recorded so that you can hear it later.

## 8. Use Every Source of Physics Help Available

If you find physics challenging, don’t make yourself alone. Try all help sources that are accessible to you. Every student deserves maximum guidance to score good grades. Yes, some options cost money, but there are plenty of free resources available. Try these:

• Try MTS online physics tuition . Our personalized and one-on-one lessons will improve your grades.
• Schedule an after-school consultation with your physics teacher.
• Use third-party resources, such as physics websites, libraries, or YouTube lectures.
• Join a physics study group.

## 9. Take Time for Revision

Are you performing poorly on practice or mock exams? It means that you are not investing time in revision. Yes, mock exams don’t hold as much weightage as the real exam, but they are excellent at recognizing weak areas.

Don’t only focus on weaknesses because all topics should be revised. Many students put revision at the end of their priorities and end up missing difficult topics. Revision actually starts after each class when you see your notes.

Tip: You can also ask your peers what revision tricks they follow.

## 10. Always Review Class Notes

Don’t be one of those students who only open their textbooks when it’s time for exam revision. Try reading the class notes before the next class begins. This habit will ensure smooth memorization of everything that has been taught during lectures.

Tip: Don’t spend too much time on note review because it will be hard to retain the concepts. Therefore, only review your class notes before the next lecture and before hitting the bed.

## 11. Make It Fun!

A huge possible reason why you are lacking as a physics student is low enthusiasm to focus and grow. Poor motivation usually happens due to the fear of doing well and the exam burden. Very few students have fun learning physics.

## 12. Keep Up the Momentum

Try to keep up, and if you fall behind, recover quickly. As stated earlier, physics spreads upwards and outwards from key theories and concepts. If you’re unaware or weak in just one topic, your course will be harder to understand.

Therefore, be very disciplined towards this subject. Always keep up with the daily study, regular practice, assigned homework, and review previous topics so that they remain fresh in your mind. Try these useful tips for improving your physics grades:

• Invest some time daily to go over older topics discussed in lectures. There are online apps available that provide automatic reviews or summaries of lessons and notes. Your daily revision time should be 15 to 30 minutes.
• Read the book chapters before the class begins and note down the main points so you are mentally prepared to learn that content. This will improve your attention and interest towards the lesson.
• If you’re weak in a topic, reach out to your nerd friend, teacher, or seniors. Don’t delay asking the question right away because whatever knowledge is built on that topic will also be unclear.
• Practice the problems in your textbook and make sure that you know them on a conceptual level. Plus, revisit those problems periodically to practice again. You can also practice problems based on the same topic and level from the internet.

## 13. Adopt a Conceptual Approach

Whether you’re learning the basics of dimensions or finding out an answer, conceptual learning will divide knowledge into digestible bits. Always be clear on the core concepts, but draw the bigger picture at the back of your mind. Merge concepts in a solid understanding of the current problem.

Don’t ever forget that physics is constructed on these conceptual building blocks. The universe and physical world’s laws are used to study all topics. Good grades are impossible to achieve without having a firm understanding of the fundamentals. Moreover, you should have enough knowledge to know how and when to apply these laws to solve a problem.

So stop having tough times with your physics course and use conceptual building blocks. Whenever you get confused, it will be easy to go back a few steps.

## 14. Have a Solid Grip on SI Units

Most students run away from the SI units, but they aren’t as complicated as they seem. These units matter more than you think. For example, if you forget an SI unit after solving a numerical or write a wrong one, it will impact your grades.

See this basic SI unit table that you should memorize:

## 15. Focus on Illustrations

Visual learning never does harm and increases productivity. Physics theories and concepts can be learned through images, graphs, doodles, or illustrations. Not only this, visual learning sticks information longer in the brain. Consider this visual, for example:

## Wrapping it Up

The above 15 tips are helpful in learning physics quickly and productively. Physics is a compulsory subject for many students all over the world, and it deserves undivided attention throughout the course. And with so much technology in today’s era, there are loads of online tools and practice tests to know your understanding. Even the students who dislike physics or find it boring will see it differently after following the above tips.

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• Non-Horizontally Launched Projectile Problems

There are two basic types of projectile problems that we will discuss in this course. While the general principles are the same for each type of problem, the approach will vary due to the fact the problems differ in terms of their initial conditions. The two types of problems are:

A projectile is launched with an initial horizontal velocity from an elevated position and follows a parabolic path to the ground. Predictable unknowns include the initial speed of the projectile, the initial height of the projectile, the time of flight, and the horizontal distance of the projectile.

Examples of this type of problem are

• A pool ball leaves a 0.60-meter high table with an initial horizontal velocity of 2.4 m/s. Predict the time required for the pool ball to fall to the ground and the horizontal distance between the table's edge and the ball's landing location.

A soccer ball is kicked horizontally off a 22.0-meter high hill and lands a distance of 35.0 meters from the edge of the hill. Determine the initial horizontal velocity of the soccer ball.

A projectile is launched at an angle to the horizontal and rises upwards to a peak while moving horizontally. Upon reaching the peak, the projectile falls with a motion that is symmetrical to its path upwards to the peak. Predictable unknowns include the time of flight, the horizontal range, and the height of the projectile when it is at its peak.

• A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the football.
• A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28-degrees above the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the long-jumper.

The second problem type will be the subject of the next part of Lesson 2 . In this part of Lesson 2, we will focus on the first type of problem - sometimes referred to as horizontally launched projectile problems. Three common kinematic equations that will be used for both type of problems include the following:

d = v i •t + 0.5*a*t 2 v f = v i + a•t v f 2  = v i 2  + 2*a•d

## Equations for the Horizontal Motion of a Projectile

The above equations work well for motion in one-dimension, but a projectile is usually moving in two dimensions - both horizontally and vertically. Since these two components of motion are independent of each other, two distinctly separate sets of equations are needed - one for the projectile's horizontal motion and one for its vertical motion. Thus, the three equations above are transformed into two sets of three equations. For the horizontal components of motion, the equations are

x = v i x •t + 0.5*a x *t 2

v f x  = v i x  + a x •t

v f x 2  = v i x 2  + 2*a x •x

Of these three equations, the top equation is the most commonly used. An application of projectile concepts to each of these equations would also lead one to conclude that any term with a x in it would cancel out of the equation since a x = 0 m/s/s . Once this cancellation of ax terms is performed, the only equation of usefulness is:

x = v i x •t

## Equations for the Vertical Motion of a Projectile

For the vertical components of motion, the three equations are

y = v iy •t + 0.5*a y *t 2

v fy  = v iy  + a y •t

v fy 2  = v iy 2  + 2*a y •y

In each of the above equations, the vertical acceleration of a projectile is known to be -9.8 m/s/s (the acceleration of gravity). Furthermore, for the special case of the first type of problem (horizontally launched projectile problems), v iy = 0 m/s. Thus, any term with v iy in it will cancel out of the equation.

The two sets of three equations above are the kinematic equations that will be used to solve projectile motion problems.

## Solving Projectile Problems

To illustrate the usefulness of the above equations in making predictions about the motion of a projectile, consider the solution to the following problem.

The solution of this problem begins by equating the known or given values with the symbols of the kinematic equations - x, y, v ix , v iy , a x , a y , and t. Because horizontal and vertical information is used separately, it is a wise idea to organized the given information in two columns - one column for horizontal information and one column for vertical information. In this case, the following information is either given or implied in the problem statement:

As indicated in the table, the unknown quantity is the horizontal displacement (and the time of flight) of the pool ball. The solution of the problem now requires the selection of an appropriate strategy for using the kinematic equations and the known information to solve for the unknown quantities. It will almost always be the case that such a strategy demands that one of the vertical equations be used to determine the time of flight of the projectile and then one of the horizontal equations be used to find the other unknown quantities (or vice versa - first use the horizontal and then the vertical equation). An organized listing of known quantities (as in the table above) provides cues for the selection of the strategy. For example, the table above reveals that there are three quantities known about the vertical motion of the pool ball. Since each equation has four variables in it, knowledge of three of the variables allows one to calculate a fourth variable. Thus, it would be reasonable that a vertical equation is used with the vertical values to determine time and then the horizontal equations be used to determine the horizontal displacement (x). The first vertical equation (y = v iy •t +0.5•a y •t 2 ) will allow for the determination of the time. Once the appropriate equation has been selected, the physics problem becomes transformed into an algebra problem. By substitution of known values, the equation takes the form of

Since the first term on the right side of the equation reduces to 0, the equation can be simplified to

If both sides of the equation are divided by -5.0 m/s/s, the equation becomes

By taking the square root of both sides of the equation, the time of flight can then be determined .

Once the time has been determined, a horizontal equation can be used to determine the horizontal displacement of the pool ball. Recall from the given information , v ix = 2.4 m/s and a x = 0 m/s/s. The first horizontal equation (x = v ix •t + 0.5•a x •t 2 ) can then be used to solve for "x." With the equation selected, the physics problem once more becomes transformed into an algebra problem. By substitution of known values, the equation takes the form of

Since the second term on the right side of the equation reduces to 0, the equation can then be simplified to

The answer to the stated problem is that the pool ball is in the air for 0.35 seconds and lands a horizontal distance of 0.84 m from the edge of the pool table.

The following procedure summarizes the above problem-solving approach.

• Carefully read the problem and list known and unknown information in terms of the symbols of the kinematic equations. For convenience sake, make a table with horizontal information on one side and vertical information on the other side.
• Identify the unknown quantity that the problem requests you to solve for.
• Select either a horizontal or vertical equation to solve for the time of flight of the projectile.
• With the time determined, use one of the other equations to solve for the unknown. (Usually, if a horizontal equation is used to solve for time, then a vertical equation can be used to solve for the final unknown quantity.)

One caution is in order. The sole reliance upon 4- and 5-step procedures to solve physics problems is always a dangerous approach. Physics problems are usually just that - problems! While problems can often be simplified by the use of short procedures as the one above, not all problems can be solved with the above procedure. While steps 1 and 2 above are critical to your success in solving horizontally launched projectile problems, there will always be a problem that doesn't fit the mold . Problem solving is not like cooking; it is not a mere matter of following a recipe. Rather, problem solving requires careful reading, a firm grasp of conceptual physics, critical thought and analysis, and lots of disciplined practice. Never divorce conceptual understanding and critical thinking from your approach to solving problems.

Use y = v iy • t + 0.5 • a y • t 2 to solve for time; the time of flight is 2.12 seconds.

Now use x = v ix • t + 0.5 • a x • t 2 to solve for v ix

Note that a x is 0 m/s/s so the last term on the right side of the equation cancels. By substituting 35.0 m for x and 2.12 s for t, the v ix can be found to be 16.5 m/s.

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## Scientists use generative AI to answer complex questions in physics

Images for download on the MIT News office website are made available to non-commercial entities, press and the general public under a Creative Commons Attribution Non-Commercial No Derivatives license . You may not alter the images provided, other than to crop them to size. A credit line must be used when reproducing images; if one is not provided below, credit the images to "MIT."

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When water freezes, it transitions from a liquid phase to a solid phase, resulting in a drastic change in properties like density and volume. Phase transitions in water are so common most of us probably don’t even think about them, but phase transitions in novel materials or complex physical systems are an important area of study.

To fully understand these systems, scientists must be able to recognize phases and detect the transitions between. But how to quantify phase changes in an unknown system is often unclear, especially when data are scarce.

Researchers from MIT and the University of Basel in Switzerland applied generative artificial intelligence models to this problem, developing a new machine-learning framework that can automatically map out phase diagrams for novel physical systems.

Their physics-informed machine-learning approach is more efficient than laborious, manual techniques which rely on theoretical expertise. Importantly, because their approach leverages generative models, it does not require huge, labeled training datasets used in other machine-learning techniques.

Such a framework could help scientists investigate the thermodynamic properties of novel materials or detect entanglement in quantum systems, for instance. Ultimately, this technique could make it possible for scientists to discover unknown phases of matter autonomously.

“If you have a new system with fully unknown properties, how would you choose which observable quantity to study? The hope, at least with data-driven tools, is that you could scan large new systems in an automated way, and it will point you to important changes in the system. This might be a tool in the pipeline of automated scientific discovery of new, exotic properties of phases,” says Frank Schäfer, a postdoc in the Julia Lab in the Computer Science and Artificial Intelligence Laboratory (CSAIL) and co-author of a paper on this approach.

Joining Schäfer on the paper are first author Julian Arnold, a graduate student at the University of Basel; Alan Edelman, applied mathematics professor in the Department of Mathematics and leader of the Julia Lab; and senior author Christoph Bruder, professor in the Department of Physics at the University of Basel. The research is published today in Physical Review Letters.

Detecting phase transitions using AI

While water transitioning to ice might be among the most obvious examples of a phase change, more exotic phase changes, like when a material transitions from being a normal conductor to a superconductor, are of keen interest to scientists.

These transitions can be detected by identifying an “order parameter,” a quantity that is important and expected to change. For instance, water freezes and transitions to a solid phase (ice) when its temperature drops below 0 degrees Celsius. In this case, an appropriate order parameter could be defined in terms of the proportion of water molecules that are part of the crystalline lattice versus those that remain in a disordered state.

In the past, researchers have relied on physics expertise to build phase diagrams manually, drawing on theoretical understanding to know which order parameters are important. Not only is this tedious for complex systems, and perhaps impossible for unknown systems with new behaviors, but it also introduces human bias into the solution.

More recently, researchers have begun using machine learning to build discriminative classifiers that can solve this task by learning to classify a measurement statistic as coming from a particular phase of the physical system, the same way such models classify an image as a cat or dog.

The MIT researchers demonstrated how generative models can be used to solve this classification task much more efficiently, and in a physics-informed manner.

The Julia Programming Language , a popular language for scientific computing that is also used in MIT’s introductory linear algebra classes, offers many tools that make it invaluable for constructing such generative models, Schäfer adds.

Generative models, like those that underlie ChatGPT and Dall-E, typically work by estimating the probability distribution of some data, which they use to generate new data points that fit the distribution (such as new cat images that are similar to existing cat images).

However, when simulations of a physical system using tried-and-true scientific techniques are available, researchers get a model of its probability distribution for free. This distribution describes the measurement statistics of the physical system.

A more knowledgeable model

The MIT team’s insight is that this probability distribution also defines a generative model upon which a classifier can be constructed. They plug the generative model into standard statistical formulas to directly construct a classifier instead of learning it from samples, as was done with discriminative approaches.

“This is a really nice way of incorporating something you know about your physical system deep inside your machine-learning scheme. It goes far beyond just performing feature engineering on your data samples or simple inductive biases,” Schäfer says.

This generative classifier can determine what phase the system is in given some parameter, like temperature or pressure. And because the researchers directly approximate the probability distributions underlying measurements from the physical system, the classifier has system knowledge.

This enables their method to perform better than other machine-learning techniques. And because it can work automatically without the need for extensive training, their approach significantly enhances the computational efficiency of identifying phase transitions.

At the end of the day, similar to how one might ask ChatGPT to solve a math problem, the researchers can ask the generative classifier questions like “does this sample belong to phase I or phase II?” or “was this sample generated at high temperature or low temperature?”

Scientists could also use this approach to solve different binary classification tasks in physical systems, possibly to detect entanglement in quantum systems (Is the state entangled or not?) or determine whether theory A or B is best suited to solve a particular problem. They could also use this approach to better understand and improve large language models like ChatGPT by identifying how certain parameters should be tuned so the chatbot gives the best outputs.

In the future, the researchers also want to study theoretical guarantees regarding how many measurements they would need to effectively detect phase transitions and estimate the amount of computation that would require.

This work was funded, in part, by the Swiss National Science Foundation, the MIT-Switzerland Lockheed Martin Seed Fund, and MIT International Science and Technology Initiatives.

• Frank Schäfer
• Alan Edelman
• Computer Science and Artificial Intelligence Laboratory
• Department of Mathematics
• Department of Electrical Engineering and Computer Science

## Related Topics

• Mathematics
• Computer science and technology
• Artificial intelligence
• Computer modeling
• Computer Science and Artificial Intelligence Laboratory (CSAIL)
• Electrical Engineering & Computer Science (eecs)

## From physics to generative AI: An AI model for advanced pattern generation

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May 16, 2024

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## Scientists use generative AI to answer complex questions in physics

by Adam Zewe, Massachusetts Institute of Technology

When water freezes, it transitions from a liquid phase to a solid phase, resulting in a drastic change in properties like density and volume. Phase transitions in water are so common most of us probably don't even think about them, but phase transitions in novel materials or complex physical systems are an important area of study.

To fully understand these systems, scientists must be able to recognize phases and detect the transitions between. But how to quantify phase changes in an unknown system is often unclear, especially when data are scarce.

Researchers from MIT and the University of Basel in Switzerland applied generative artificial intelligence models to this problem, developing a new machine-learning framework that can automatically map out phase diagrams for novel physical systems.

Their physics-informed machine-learning approach is more efficient than laborious, manual techniques which rely on theoretical expertise. Importantly, because their approach leverages generative models, it does not require huge, labeled training datasets used in other machine-learning techniques.

Such a framework could help scientists investigate the thermodynamic properties of novel materials or detect entanglement in quantum systems, for instance. Ultimately, this technique could make it possible for scientists to discover unknown phases of matter autonomously.

"If you have a new system with fully unknown properties, how would you choose which observable quantity to study? The hope, at least with data-driven tools, is that you could scan large new systems in an automated way, and it will point you to important changes in the system.

"This might be a tool in the pipeline of automated scientific discovery of new, exotic properties of phases," says Frank Schäfer, a postdoc in the Julia Lab in the Computer Science and Artificial Intelligence Laboratory (CSAIL) and co-author of a paper on this approach.

Joining Schäfer on the paper are first author Julian Arnold, a graduate student at the University of Basel; Alan Edelman, applied mathematics professor in the Department of Mathematics and leader of the Julia Lab; and senior author Christoph Bruder, professor in the Department of Physics at the University of Basel.

The research is published in Physical Review Letters .

## Detecting phase transitions using AI

While water transitioning to ice might be among the most obvious examples of a phase change, more exotic phase changes, like when a material transitions from being a normal conductor to a superconductor, are of keen interest to scientists.

These transitions can be detected by identifying an "order parameter," a quantity that is important and expected to change. For instance, water freezes and transitions to a solid phase (ice) when its temperature drops below 0°C. In this case, an appropriate order parameter could be defined in terms of the proportion of water molecules that are part of the crystalline lattice versus those that remain in a disordered state.

In the past, researchers have relied on physics expertise to build phase diagrams manually, drawing on theoretical understanding to know which order parameters are important. Not only is this tedious for complex systems, and perhaps impossible for unknown systems with new behaviors, but it also introduces human bias into the solution.

More recently, researchers have begun using machine learning to build discriminative classifiers that can solve this task by learning to classify a measurement statistic as coming from a particular phase of the physical system, the same way such models classify an image as a cat or dog.

The MIT researchers demonstrated how generative models can be used to solve this classification task much more efficiently, and in a physics-informed manner.

The Julia Programming Language , a popular language for scientific computing that is also used in MIT's introductory linear algebra classes, offers many tools that make it invaluable for constructing such generative models, Schäfer adds.

Generative models, like those that underlie ChatGPT and Dall-E, typically work by estimating the probability distribution of some data, which they use to generate new data points that fit the distribution (such as new cat images that are similar to existing cat images).

However, when simulations of a physical system using tried-and-true scientific techniques are available, researchers get a model of its probability distribution for free. This distribution describes the measurement statistics of the physical system.

## A more knowledgeable model

The MIT team's insight is that this probability distribution also defines a generative model upon which a classifier can be constructed. They plug the generative model into standard statistical formulas to directly construct a classifier instead of learning it from samples, as was done with discriminative approaches.

"This is a really nice way of incorporating something you know about your physical system deep inside your machine-learning scheme. It goes far beyond just performing feature engineering on your data samples or simple inductive biases," Schäfer says.

This generative classifier can determine what phase the system is in given some parameter, like temperature or pressure. And because the researchers directly approximate the probability distributions underlying measurements from the physical system, the classifier has system knowledge.

This enables their method to perform better than other machine-learning techniques. And because it can work automatically without the need for extensive training, their approach significantly enhances the computational efficiency of identifying phase transitions .

At the end of the day, similar to how one might ask ChatGPT to solve a math problem, the researchers can ask the generative classifier questions like "does this sample belong to phase I or phase II?" or "was this sample generated at high temperature or low temperature?"

Scientists could also use this approach to solve different binary classification tasks in physical systems, possibly to detect entanglement in quantum systems (Is the state entangled or not?) or determine whether theory A or B is best suited to solve a particular problem. They could also use this approach to better understand and improve large language models like ChatGPT by identifying how certain parameters should be tuned so the chatbot gives the best outputs.

In the future, the researchers also want to study theoretical guarantees regarding how many measurements they would need to effectively detect phase transitions and estimate the amount of computation that would require.

Journal information: Physical Review Letters , arXiv

Provided by Massachusetts Institute of Technology

This story is republished courtesy of MIT News ( web.mit.edu/newsoffice/ ), a popular site that covers news about MIT research, innovation and teaching.

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## Android's Circle to Search can now help students solve math and physics homework

It can show step-by-step instructions on how to solve a range of math and physics problems..

Google has introduced another capability for its Circle to Search feature at the company's annual I/O developer conference, and it's something that could help students better understand potentially difficult class topics . The feature will now be able to show them step-by-step instructions for a "range of physics and math word problems." They just have to activate the feature by long-pressing the home button or navigation bar and then circling the problem that's got them stumped, though some math problems will require users to be signed up for Google's experimental Search Labs feature.

The company says Circle to Search's new capability was made possible by its new family of AI models called LearnLM that was specifically created and fine-tuned for learning. It's also planning to make adjustments to this particular capability and to roll out an upgraded version later this year that could solve even more complex problems "involving symbolic formulas, diagrams, graphs and more." Google launched Circle to Search earlier this year at a Samsung Unpacked event, because the feature was initially available on Galaxy 24, as well as on Pixel 8 devices. It's now also out for the Galaxy S23, Galaxy S22, Z Fold, Z Flip, Pixel 6 and Pixel 7 devices, and it'll likely make its way to more hardware in the future.

In addition to the new Circle to Search capability, Google has also revealed that devices that can support the Gemini for Android chatbot assistant will now be able to bring it up as an overlay on top of the application that's currently open. Users can then drag and drop images straight from the overlay into apps like Gmail, for instance, or use the overlay to look up information without having to swipe away from whatever they're doing. They can tap "Ask this video" to find specific information within a YouTube video that's open, and if they have access to Gemini Advanced, they can use the "Ask this PDF" option to find information from within lengthy documents.

Catch up on all the news from Google I/O 2024 right here !

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## Experience Google AI in even more ways on Android

May 14, 2024

By building AI right into the Android operating system, we're reimagining how you can interact with your phone.

• Bullet points
• Circle to Search gets smarter, helping students solve physics and math problems directly from their phones and tablets.
• Gemini on Android improves context understanding, allowing users to drag and drop generated images and ask questions about videos and PDFs.
• Gemini Nano with Multimodality coming to Pixel, bringing multimodal capabilities for richer image descriptions and scam alerts during phone calls.
• Android 15 and ecosystem updates coming tomorrow.
• Basic explainer

Google is making Android phones smarter with AI.

Circle to Search can now help students with homework.

Android phones will soon be able to alert you to suspected scams during phone calls.

## Explore other styles:

We’re at a once-in-a-generation moment where the latest advancements in AI are reinventing what phones can do. With Google AI at the core of Android’s operating system, the billions of people who use Android can now interact with their devices in entirely new ways.

Today, we’re sharing updates that let you experience Google AI on Android.

## Circle to Search can now help students with homework

With Circle to Search built directly into the user experience, you can search anything you see on your phone using a simple gesture — without having to stop what you’re doing or switch to a different app. Since launching at Samsung Unpacked , we’ve added new capabilities to Circle to Search, like full-screen translation , and we’ve expanded availability to more Pixel and Samsung devices.

Starting today, Circle to Search can now help students with homework, giving them a deeper understanding, not just an answer — directly from their phones and tablets. When students circle a prompt they’re stuck on, they’ll get step-by-step instructions to solve a range of physics and math 1 word problems without leaving their digital info sheet or syllabus. Later this year, Circle to Search will be able to help solve even more complex problems involving symbolic formulas, diagrams, graphs and more. This is all possible due to our LearnLM effort to enhance our models and products for learning.

Circle to Search is already available on more than 100 million devices today. With plans to bring the experience to more devices, we’re on track to double that by the end of the year.

## Gemini will get even better at understanding context to assist you in getting things done

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Soon, you’ll be able to bring up Gemini's overlay on top of the app you're in to easily use Gemini in more ways. For example, you can drag and drop generated images into Gmail, Google Messages and other places, or tap “Ask this video” to find specific information in a YouTube video. If you have Gemini Advanced, you’ll also have the option to “Ask this PDF” to quickly get answers without having to scroll through multiple pages. This update will roll out to hundreds of millions of devices over the next few months.

And we’ll continue to improve Gemini to give you more dynamic suggestions related to what’s on your screen.

## Full multimodal capabilities coming to Gemini Nano

Android is the first mobile operating system that includes a built-in, on-device foundation model. With Gemini Nano, we’re able to bring experiences to you quickly and keep your information completely private to you. Starting with Pixel later this year, we’ll be introducing our latest model, Gemini Nano with Multimodality. This means your phone will not just be able to process text input but also understand more information in context like sights, sounds and spoken language.

## Clearer descriptions with TalkBack

Later this year, Gemini Nano’s multimodal capabilities are coming to TalkBack, helping people who experience blindness or low vision get richer and clearer descriptions of what’s happening in an image. On average, TalkBack users come across 90 unlabeled images per day. This update will help fill in missing information — whether it’s more details about what’s in a photo that family or friends sent or the style and cut of clothes when shopping online. Since Gemini Nano is on-device, these descriptions happen quickly and even work when there's no network connection.

According to a recent report , in a 12-month period, people lost more than \$1 trillion to fraud. We’re testing a new feature that uses Gemini Nano to provide real-time alerts during a call if it detects conversation patterns commonly associated with scams. For example, you would receive an alert if a “bank representative” asks you to urgently transfer funds, make a payment with a gift card or requests personal information like card PINs or passwords, which are uncommon bank requests . This protection all happens on-device, so your conversation stays private to you. We’ll share more about this opt-in feature later this year.

## More to come on Android

We’re just getting started with how on-device AI can change what your phone can do, and we’ll continue building Google AI into every part of the smartphone experience with Pixel, Samsung and more. If you’re a developer, check out the Android Developers blog to learn how you can build with our latest AI models and tools, like Gemini Nano and Gemini in Android Studio.

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Done. Just one step more.

You can also subscribe with a different email address .

Available for some math word problems when opted into Search Labs.

## Android’s theft protection features keep your device and data safe

Let’s stay in touch. Get the latest news from Google in your inbox.

1. How to Solve Any Physics Problem: 10 Steps (with Pictures)

Calm down. It is just a problem, not the end of the world! 2. Read through the problem once. If it is a long problem, read and understand it in parts till you get even a slight understanding of what is going on. 3. Draw a diagram. It cannot be emphasized enough how much easier a problem will be once it is drawn out.

2. PDF An Expert's Approach to Solving Physics Problems

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5. 1.7 Solving Problems in Physics

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8. 1.4: Solving Physics Problems

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27. Scientists use generative AI to answer complex questions in physics

The MIT researchers demonstrated how generative models can be used to solve this classification task much more efficiently, and in a physics-informed manner. The Julia Programming Language , a popular language for scientific computing that is also used in MIT's introductory linear algebra classes, offers many tools that make it invaluable for ...

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