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AP Physics 1: Exam Prep13 chapters | 143 lessons | 6 flashcard sets

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Lesson Transcript

Instructor:
*Damien Howard*

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson, you'll learn how to solve one-dimensional elastic collision problems. You'll find that understanding the conservation of momentum and conservation of kinetic energy is essential to solving these types of problems.

When you think about a collision, what normally comes to mind? For many people, it will recall two vehicles crashing into each other. Whether watching a TV show of cars crashing while in a race or seeing the aftermath of an accident on the road, you've probably had some first-hand experience in seeing a type of collision.

A couple of different things can happen when two cars collide. They can bounce off each other, or they can stick together. We call these two categories of collisions elastic and inelastic collisions respectively, and these categories apply to more than just car crashes. Any collision where two things bounce off one another, like a bowling ball hitting pins, is called an **elastic collision**. Similarly, any collision where two things stick together, like one football player tackling another, is considered an **inelastic collision**. In this lesson, we'll focus on the former and dive into learning about the physics that occur in a one-dimensional elastic collision.

While working on collisions in introductory physics courses, what we're really doing is exploring conservation laws. Specifically, we look at how the conservation of momentum and kinetic energy work in a collision.

When we say momentum is conserved in an elastic collision, **conservation of momentum** in other words, what we really mean is that the total momentum of the two objects colliding is the same before and after the collision. Mathematically, we write this out as follows:

Here * p* stands for momentum,

To see how we use this formula in practice, let's work through an example problem together. When children play a game of marbles, their goal is to roll a large marble at smaller ones and bounce them out of a boundary drawn on the ground. One child rolls their large marble with a mass of 0.005 kg at a velocity of 0.6 m/s. It strikes a stationary marble with a mass of 0.0036 kg. After the collision, the large marble continues forward with a velocity of 0.2 m/s. What is the final velocity of the smaller marble?

With the information given, we know the mass of both marbles (*m1*, *m2*), their initial velocities (* vi1*,

*m1*** vi1* +

(0.005 kg)(0.6 m/s) + (0.0036 kg)(0 m/s) = (0.005 kg)(0.2 m/s) + (0.0036 kg)(* vf2*)

As you can see, we gave the smaller marble an initial velocity of 0 m/s. This is because before the collision it was standing still, meaning it had no velocity.

0.003 kg m/s + 0 kg m/s = 0.001 kg m/s + (0.0036 kg)(* vf2*)

0.003 kg m/s = 0.001 kg m/s + (0.0036 kg)(* vf2*)

0.002 kg m/s = (0.0036 kg)(* vf2*)

0.56 m/s = **v**f2

One big difference between inelastic and elastic collisions is that kinetic energy is only conserved in the latter. In fact, a collision must be perfectly elastic for the conservation of energy to occur. If there is any inelasticity at all, then some energy is lost.

When a collision is perfectly elastic, the law of **conservation of kinetic energy** tells us that the total kinetic energy of the colliding objects is the same before and after the collision. We write this out mathematically much like we did with the conservation of momentum:

In these equations * KE* stands for kinetic energy, and the rest of the variables and subscripts are the same as they were in the conservation of momentum equations.

When comparing the final formula above to what we have for the conservation of momentum, you can see they're actually very similar. If we worked through an example problem it would be much like the one in the previous section, but what we can do instead is use the conservation of momentum and kinetic energy together to solve a more complex problem.

Two bumper cars at an amusement park drive directly toward each other. The first bumper car has a mass (*m1*) of 200 kg and is travelling with an initial velocity (* vi1*) of 4 m/s. The second bumper car has a mass (

There are quite a few steps involved to solve this problem, so let's work through them one at a time. The first thing you need to do is take the formula for the conservation of kinetic energy and rearrange it so all *m1* factors are located on one side of the equation, and *m2* factors are on the other side.

Next, you'll need to factor out your squares on each side of the equation.

Once you've got this done, you'll want to switch over to the formula for the conservation of momentum and get the mass factors on either side of the equation just like you did before.

You'll then need to divide your factored out equation for the conservation of kinetic energy by your factored out formula for the conservation of momentum.

Now, we will rearrange this equation to put * vf2* by itself on one side of the equals sign.

This equation can now be plugged back into the original formula for the conservation of momentum to solve for * vf1*.

At this point, you'll want to get the * vf1* turn itself on one side of the equals sign.

At last, we're ready to insert the information from our problem to solve for the first final velocity.

Finally, now that we know one of the final velocities, we can plug back its value into the original formula for the conservation of momentum to solve for the second final velocity.

*m1*** vi1* +

(200 kg)(4 m/s) + (160 kg)(-6 m/s) = (200 kg)(-4.89 m/s) + (160 kg)(* vf2*)

800 kg m/s - 960 kg m/s = -978 kg m/s + (160 kg)(* vf2*)

-160 kg m/s = -978 kg m/s + (160 kg)(* vf2*)

818 kg m/s = (160 kg)(* vf2*)

5.11 m/s = **v**f2

When two objects collide, one of two types of collisions can occur. In an **elastic collision**, the objects bounce off each other after colliding, while in an **inelastic collision**, the objects that collide stick to each other after the collision. When solving one-dimensional collision problems in a physics course, what we're really working with are conservation laws. For an elastic collision, there are two of these conservation laws that apply.

The first is the **conservation of momentum**, and it states that the total momentum before and after the collision must be the same:

Unlike the conservation of momentum, the second conservation law applies only to elastic collisions, and not inelastic ones. It is the **conservation of kinetic energy**, which tells us that the total kinetic energy must be the same before and after the collision.

You'll use these two conservation equations constantly when working on problems that involve one-dimensional elastic collisions.

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AP Physics 1: Exam Prep13 chapters | 143 lessons | 6 flashcard sets

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