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Impulse Comparison for Elastic and Inelastic Collisions

Keith Michaelsen, Southington High School, Southington, CT, contacted us to discuss ways to show students that the impulse delivered during an elastic collision is twice the impulse delivered by an inelastic collision. This is a counterintuitive concept, and performing an experiment to observe this can be a challenge. The main issues are:

  1. To compare the change in momentum between the two different types of collisions, you have to perform the experiment multiple times, and for each trial you need to have a consistent initial momentum.
  2. Friction must be minimized, but that can be accomplished with a low-friction cart. Producing an elastic collision with a low-friction cart is fairly easy since we could use magnets or springs during the collision. For example, a cart could move along a track and collide with a spring at end of the track. Producing perfectly inelastic collisions can be difficult, because objects tend to bounce.

We tried different combinations of equipment and different experimental setups, and came up with a nice experiment. We ended up using a Dual-Range Force Sensor, a Vernier Photogate, our Dynamics System, and our Bumper and Launcher Kit.

On one end of the track, we placed an end stop. We attached a hoop spring from the Bumper and Launcher Kit to the force sensor and laid the force sensor in the track, butted up against the end stop. We attached a photogate to the track and attached a picket fence to the cart.

On the other end of the track we placed the track bracket from the Bumper and Launcher Kit, and we attached the other hoop spring to the bracket. We launched the cart from a consistent compression of the hoop spring. This produced a consistent velocity.

Elastic and Inelastic Setup 1
Elastic and Inelastic Setup 2

When the cart was launched toward the other end, it collided and rebounded from the hoop bumper on the force sensor. This generated a force vs. time graph. We used a data-collection rate of 500 samples/second and collected 10 seconds of data. We set the photogate to work in the Gate mode. We used the integral feature of Logger Pro to determine the impulse and collected five trials of data for each collision type.

To collect data during an inelastic collision, we replaced the hoop bumper with a piece of clay rolled into a cone. We also put a small piece of clay on the front of the cart. We then launched it by pulling it back the same amount. When the cart collided on the other end, it stuck to the clay. We collected five runs.

Elastic and Inelastic Setup 3

The following graphs show a comparison of the impulse from two runs—one elastic and the other inelastic. The scaling of the graphs is the same so that you can see the difference in the collisions. The elastic collision shows a longer interaction time and a smaller maximum force. The inelastic collision (the graph on the bottom) displayed some interesting results. The difficulty in performing this experiment is using materials that produce an inelastic collision. We felt that clay was a good choice, however, the bottom graph shows us that the cart is bouncing back very slightly. As a result, when we used the Logger Pro integral tool, we selected the region from the beginning of the collision until the force returned to zero and the cart had come to a complete stop.

Elastic and Inelastic Collision Graphs

When we analyzed the data over the five runs, we obtained the following results. The change in momentum was calculated from the mass of the cart and the change in velocity. The impulse is found using the integral of the force vs. time graph.

Trial Change in Momentum (kg m/s) Impulse (N s)
Elastic Collision 0.32 0.32
Inelastic Collision 0.17 0.18

The impulse values determined through the velocity-change calculation and the force integral were consistent. The impulse from the elastic collision was very close to twice the impulse of the inelastic collision. This is the result that we sought. The elastic impulse is a little less than twice the inelastic impulse. That the ratio is just under 2 could be due to the fact that the “elastic” collision is losing some energy. We saw this as a slightly smaller speed after the impulse. Perhaps a magnetic bumper would see a more nearly elastic collision, and yield a ratio closer to 2.

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