# Collisions

They happen all around us. Raindrops fall from the sky and collide with the ground. We have all heard of black holes colliding in spacetime. More often than you’d like, random furniture collides with your harmless toes (your furniture might beg to differ). The last one might have left you wanting to cushion your furniture all over. But why? Why does colliding with a cushion not hurt, while wood gives you a stubbed toe?

Understanding collisions has a rather curious history — a brief account is presented in Cassidy, Holton, and Rutherford, Understanding Physics, Section 5.6. The world understood collisions through Newton’s Laws of motion, specifically, the second law:

which means that the momentum ℗ of a particle changes with time (t) by the amount exactly equal to the amount of Force (F) applied on the particle. For a system of particles, this law translates to

meaning that the time rate of change of total momentum of the system ($$P_{sys}$$) is equal to the net “external” force ($$F_{ext}$$) applied on it.

During collisions, particles fall towards each other due to two reasons. One is due to forces between them or internal forces. Two is inertia when a body is set to move on its own towards another particle. For instance, the collision between a raindrop and the Earth is because of the gravitational force between them, and the furniture with your toe is due to your toe’s inertia. Collisions don’t *need* external forces to be applied. Therefore, for collisions, we would set $$F_{ext}$$$$=0$$, which means that $$\dfrac{dP_{sys}}{dt}$$$$=0$$.

So, the system’s momentum ($$P_{sys}$$) remains constant over time. In other words, $$P_{sys}$$ is conserved. This is the **Law of Conservation of Momentum**.

To see momentum conservation in collisions, some students conducted simple experiments during a session I took for their General Physics course on 9 February 2023. For our experiments, we used metal balls of various sizes (see Fig. 1.) and taped some rulers to the tables to create tracks for the balls to roll on. We split the class into small groups to set the ball rolling (pun intended). Before performing each experiment, I asked the groups to note their expectation of the outcome. Then, I asked them to experiment and see if it differed from the actual result and justify it. This led to some interesting hypotheses and further experiments to test them!

### Experiments

The first experiment we tried was colliding a ball at rest ($$b_1$$) with another identical ball ($$b_2$$) rolled towards the resting ball ($$b_1$$).

*Vid. 1. Colliding a resting ball with a rolling ball.*

### Expectation

All groups unanimously expected that $$b_2$$ would stop immediately after the collision and $$b_1$$ would roll at the same speed that $$b_2$$ had before the collision.

This expectation is fairly intuitive but very interesting because we do not imagine that $$b_2$$ rolls backwards at a hundred times its initial speed and $$b_1$$ shoots forward with hundred-and-one times the speed of $$b_2$$ The momentum would still be conserved here because

(Here $${m}$$ is the mass of each of the identical balls and $${v}$$ is the initial velocity of $$b_2$$.)

This implies there should be infinitely many possible outcomes, like ($$2001v$$,$$-2000v$$), ($$101v$$,$$-100v$$), etc. But we expect the outcome to be $$(v,0)$$, where the first value in the ordered pair is the velocity of $$b_1$$ and the second value is the velocity of $$b_2$$. In fact, it would be funny (weird and scary) if the balls shot like bullets in opposite directions after the collision (while still obeying momentum conservation). Thankfully our world does not work that way, and there should be something in addition to momentum conservation that manages to rule out all these crazy possibilities!

The system requires enormous energy to shoot the balls at bullet speeds. But our system started out with little energy and cannot create this energy on its own. Therefore, it must carry out its entire motion with the same energy with which it started. So, the total energy of the system should remain unchanged throughout its motion. This is the **Law of Conservation of Energy**.

We are interested in Kinetic Energy, which is due to the motion of each ball in our system. The Kinetic Energy of each ball clearly changes as their initial and final velocities are different. However, the *total* Kinetic Energy of the system remains unchanged.

Let us check what happens in the ($$101v$$,$$-100v$$) case:

Evidently, $$KE_{final}$$>$$KE_{initial}$$; The system should have gained $$20200$$ times its initial Kinetic Energy. But no energy is supplied to the system during the process, so this can’t happen.

Now consider the outcome that we all expected, that is, $$(v,0)$$. It is straightforward to see that $$KE_{final}=KE_{initial}$$ in this case. Any other case with factors before $${v}$$, although they may not change the momentum, will affect the KE because of the square coming from “$$v^2$$″. So all these possibilities are ruled out. Therefore this can be the only possible outcome.

*Vid. 2. Two identical balls at rest collide with one rolling ball. After collision only one ball moves out.*

In Vid. 2, two resting balls collide with one rolling ball. The result is that only one ball rolls out with the initial velocity of the rolling ball after the collision. Possibilities like only two resting balls rolling forward with $$\dfrac{v}{2}$$ velocity each or all three balls moving with $$\dfrac{v}{3}$$ velocity each, although not ruled out by momentum conservation, are ruled out by energy conservation because the velocity factors square. Calculate and check this.

Don’t play Vid. 3! What do you think would be the experiment’s outcome based on the video’s caption? Once you have guessed the outcome, play the video and see if it matches your guess.

*Vid. 3. Two balls in motion colliding with four balls at rest.*

If it does, congratulations, you got the hang of it! If not, move back to the beginning of the “Experiments” section and work through it again.

*Vid. 4. One ball in motion colliding with four balls at rest.*

I wont bother to explain this one in Vid. 4. Now you already know the secrets of the universe.

But the story hasn’t ended…

### Reality

If we carefully observe Vid. 1. (also the other videos), we will notice that the rolling ball didn’t immediately stop after the collision as we expected. ** Doomed**. Or are we? Let us think about it. Why did $$b_2$$ continue to roll after the collision?

One interesting hypothesis in the air was that the rolling ball could be slightly heavier ($${M}$$) than the resting ball ($${m}$$<$${M}$$) although they “look” identical. So the ball might have delivered $${mv}$$ to the resting ball and moved forward with the leftover momentum (($${M}$$-$${m}$$)$${v}$$) after the collision. The students devised a simple test for this “asymmetry” hypothesis. Roll the light ($${m}$$) ball while keeping $${M}$$ at rest. The rolling ball should have come to rest immediately if the hypothesis was correct. But we saw that it didn’t matter which ball we rolled. The outcome was the same. So the hypothesis was wrong.

Clearly, this system was not isolated from external forces $${-}$$ there was friction on the balls from contact with the track. So $$F_{ext}\neq 0$$ in our experimental set-up. As a result, momentum is not conserved! But if there was significant friction, the rolling ball would have all the more reason to stop immediately after the collision. So let us say the friction was negligible, that is, $$F_{ext}\approx 0$$, so the momentum is approximately conserved.

For instance, consider the case where the outcome is ($$\dfrac v{10}$$, $$\dfrac{9v}{10}$$) (Tracker can be used to determine these velocities accurately). The momentum is conserved here because $${m}$$$$\dfrac v{10}$$+$${m}$$$$\dfrac{9v}{10}$$=$${mv}$$. But the Kinetic Energy is not because of the squaring. In fact, the final Kinetic energy would be

So $$KE_{final}$$<$$KE_{initial}$$. The system had to lose $$\dfrac{18}{100}$$ fraction of $$KE_{initial}$$ for this to happen. Where did this energy go? Largely into slightly deforming the balls and heating up the system’s components.

We can now answer our original question on collisions: why does cushion not hurt while wood does? Because in wood’s case, most of the energy is delivered to your toe, deforming it. In a cushion’s case, most of the energy is lost in deforming the cushion, keeping your toe intact.

#### Acknowledgements

I am grateful to Anish for the opportunity to conduct this session with his General Physics class. I thank our Research Assistant, Pavan, for helping me gather the materials for our experiments and collecting the experiments’ videos. A big thanks to Keerthan and Anushree for kindly sharing the videos of their experiments, which I used in this post. Finally, I thank Murthy for his input to improve the post.

#### Credits

Image Credits: T Sanjana, Academic Associate in Physics, Azim Premji University

Video Credits: Keerthan and Anushree, 4^{th} year B.Sc. B.Ed. Mathematics, Azim Premji University

**Note:** The image and the videos on the page have been edited by the author for presentation purposes.