# Conservation of Kinetic Energy

Instructor: Matthew Bergstresser
Kinetic energy is the energy of motion. In this lesson we will investigate how kinetic energy is sometimes conserved and sometimes not conserved based on the type of collisions between masses.

## Kinetic Energy

When anything is moving it possesses kinetic energy (KE). This energy is determined by the product of one-half of an object's mass (m), and the square of its velocity (v) as shown in Equation 1.

Moving things are sometimes involved in collisions. Whenever the word collision is used you need to immediately think of the conservation of momentum. Momentum (p) is the product of the mass and velocity of an object (Equation 2), and momentum is always conserved in a collision as long as no outside forces are acting on the system.

Total energy is always conserved in a collision, but kinetic energy is not always conserved. This means that the total kinetic energy before the collision is not the same as the total energy after the collision. The following can happen when masses collide depending on the materials colliding.

1. The masses in the collision can bounce off each other without deforming.

2. The masses can deform and bounce off of each other after the collision.

3. The masses can stick together after the collision.

We will now discuss these three types of outcomes of a collision in terms of kinetic energy conservation.

### Elastic Collision

An elastic collision is one where very little or no kinetic energy is lost in the collision. This is generally the case where masses collide and bounce off of each other with no deformation. In elastic collisions both the conservation of momentum and the conservation of kinetic energy apply. Perfectly elastic collisions are not common in everyday life. To witness an example of a perfectly elastic collision, a special laboratory demonstration can be done whereby a frictionless track holds two carts with heavy masses. On the ends of the masses are powerful magnets with the same orientation. When the carts get close enough to each other, the magnets will cause the carts to bounce off of each other after barely touching or by not touching at all. Figure 1 illustrates this collision, with North ends (N) of magnets in each mass.

An extremely small amount of kinetic energy is lost in this collision.

An example of an elastic collision that is common in every day life is the collision of pool balls. When the balls collide they bounce off of each other, but they lose some kinetic energy to sound energy and heat energy. You can hear the collision, and if you had a sensitive enough thermometer you would detect an increase in the temperature of the pool balls. This energy came from the kinetic energy of the pool balls. That said, the amount of kinetic energy lost is so small compared to the total kinetic energy of the system that the loss can be considered to be negligible, but it is larger than in the situation where the masses with magnets collide.

Let's go through a scenario showing the conservation of kinetic energy in an elastic collision.

#### Example - Elastic Collision

Prompt: A mass M1 = 5 kg traveling at 0.5 m/s, collides with a mass M2 = 3 kg traveling at -0.25 m/s. Show that the kinetic energy of the system is conserved after the collision.

Solution: We start this solution with determining the final velocities of the masses after the collision using the center-of-mass (CM) approach. Any velocities with the apostrophe are velocities relative to the center of mass, with subscripts of f meaning final conditions, and subscripts of o meaning initial conditions.

Now we can show that kinetic energy is conserved.

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