Elastic Collisions in One Dimension

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  • 0:04 Elastic vs Inelastic…
  • 0:59 Conservation of Momentum
  • 4:19 Conservation of Kinetic Energy
  • 8:10 Lesson Summary
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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.

Elastic Collisions

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.

Conservation of Momentum

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:

conservation of momentum

Here p stands for momentum, m for mass, and v for velocity. The subscripts tell us whether we're talking about the total (or tot) amount of the momentum, the initial (i) or final (f) momentum or velocity, and the momentum, mass, or velocity of object one (1) or object two (2). In this way pitot is the initial total momentum, vf2 is the final velocity of the second object, and so forth.

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, vi2), and the final velocity of the large marble (vf1). With this information we have everything we need to find the final velocity of the small marble (vf2) using the conservation of momentum:

m1*vi1 + m2*vi2 = m1*vf1 + m2*vf2

(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 = vf2

Conservation of Kinetic Energy

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:

conservation of kinetic energy

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.

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