# Conservation of Energy in Elastic and Inelastic Collision of Objects

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### The Idea

When objects collide, they either bounce off each other or they stick together.

### What You Need

• 2 low-friction carts
• Velcro or duct tape
• low-friction track (optional)
• motion sensor
• index card
• tape

### Method

1. Measure the mass of each of the two carts.
2. Attach Velcro to the end of each of the carts, so when they meet, they stick together. You could also use duct tape formed into a loop and attached sticky-side out to each of the carts.
3. Set up the motion sensor at one end of the table.
4. Place the first cart near the motion sensor. The Velcro side should be in front, away from the motion sensor. If you have a low-friction track, place the cart on the track in a line pointing away from the motion sensor.
5. Place the second cart near the midpoint of the table with the Velcro in the rear.
6. It may be helpful to attach an index card to the back of the first cart to make it easier for the motion sensor to pick it up. If you can get away without doing this, you can avoid air resistance that could slightly affect your result.
7. Set up the motion sensor to read distance and velocity versus time.
8. Start the motion sensor. It should be on the cart setting and focused on the card of the first cart.
9. Give the first cart a push in the direction of the second cart. It should be slow enough to get a good reading from the motion sensor, but fast enough to rear-end the second cart and push it along for at least a few seconds or more. The first cart collides with the second totally inelastically, which means they stick together after the collision.
10. When both carts stop moving, stop collecting data from the motion sensor.
11. From the motion-sensor graphs, find the velocity of the first cart before the collision and the velocity of both carts joined together after the collision. The graph of velocity versus time may be a little erratic right after the collision, reflecting the impact. Pick a point where the velocity has settled down.
12. Momentum is defined as mass times velocity. Compare the momentum before and after the collision.
13. Kinetic energy is defined as ½ times the mass times the velocity squared. Compare the kinetic energy before and the kinetic energy after the collision.

The experimental setup is shown in Figure 53-1. (This actually shows a motion sensor at both ends. The previous procedure uses only one motion sensor, but this can easily extended to include both carts in motion. For simplicity, we will start out with one of the carts stationary.)

### Expected Results

The momentum of both carts before the collision should equal the momentum of both carts after the collision.

Before the collision, one of the carts is stationary, which means it has no momentum, so the moving cart is the only one with momentum before the collision.

After the collision, both carts stick together and move off with the same velocity. The combined mass of the two carts together times their combined velocity is the momentum after the collision.

Figure 53-2 shows the position versus the time graph before and after the collision obtained by a motion sensor. Notice how the slope of the line abruptly drops, indicating the collision.

Figure 53-3 shows the velocity versus time graph before and after the collision. The velocity before and after can be determined directly from the graph. You can notice a slight downward slope indicating some slowing of the carts due to friction. This is not a showstopper for the experiment, but it shows the extent to which an air track can improve the overall results.

The most reliable velocity measurement is immediately before the collision. The collision shows some bouncing around and variability in the velocity for a short period until the two carts stick together and move as one. This provides some insight into the nature of inelastic collisions, which result in the loss of kinetic energy (but not linear momentum). The most reliable postcollision velocity to use is the point where a new horizontal line begins. The results should be fairly accurate, but some losses due to friction may be encountered without an air track. Also, excessive mass can load down the wheel bearing and increase the losses to friction.

### Why It Works

The total momentum before an inelastic collision equals the total momentum after.

However, unlike an elastic collision, the kinetic energy for an inelastic collision is less after the collision.

### Other Things to Try

1. If you have two motion sensors, you can repeat this with both carts initially in motion. You can get the velocity of each cart right before the carts collide and the velocity of both carts together after the collision. Both sensors will read a positive velocity before the collision as the distance from the sensor increases. After the collision, one sensor reads a positive velocity, while the other reads a negative velocity. Results are shown in Figure 53-4.
2. Compare elastic and inelastic collisions using so-called "happy/sad" balls. The balls appear to be completely identical. However, one is elastic and bounces back from the floor, while the other is inelastic and doesn't bounce at all.
3. Hang elastic and inelastic balls to form a pendulum. Stand up a wooden block in front of the pendulum. Swing the inelastic ball first, so it doesn't quite knock the block over. Compare that to what happens with the elastic ball swung under the same conditions. Elastic collisions result in double the change in momentum as an inelastic collision under the same conditions. This is because in the elastic collision, the momentum not only stops (as it does in the case of an inelastic collision), but it also reverses itself in the other direction.

### The Point

In an elastic collision, the objects bounce off each other in such as way that linear momentum and kinetic energy are conserved. This is true to the extent that no external force occurs during the collision.

In an inelastic collision, the objects interact in such a way that linear momentum is conserved (as long as no forces affect the collision). However, kinetic energy is not conserved in an inelastic collision. In a perfectly inelastic collision, the objects stick together and move after the collision as if they were a single object.