What Are the Physics of a Car Collision?

The Distinction Between Energy and Force Can Be Very Subtle But Important.

Crashed car
Lee Haywood/Flickr/CC BY-SA 2.0

Why is it that a head-on collision between two moving vehicles is said to result in more injuries than driving a car into a wall? How do the forces felt by the driver and the energy generated differ? Focusing on the distinction between force and energy can help understand the physics involved.

Force: Colliding With a Wall

Consider case A, in which car A collides with a static, unbreakable wall. The situation begins with car A traveling at a velocity v and it ends with a velocity of 0.

The force of this situation is defined by Newton's second law of motion. Force equals mass times acceleration. In this case, the acceleration is (v - 0)/t, where t is whatever time it takes car A to come to a stop.

The car exerts this force in the direction of the wall, but the wall (which is static and unbreakable) exerts an equal force back on the car, per Newton's third law of motion. It is this equal force which causes cars to accordion up during collisions.

It is important to note that this is an idealized model. In case A, the car slams into the wall and comes to an immediate stop, which is a perfectly inelastic collision. Since the wall doesn't break or move at all, the full force of the car into the wall has to go somewhere. Either the wall is so massive that it accelerates/moves an imperceptible amount or it doesn't move at all, in which case the force of the collision actually acts on the entire planet - which is, obviously, so massive that the effects are negligible.

Force: Colliding With a Car

In case B, where car A collides with car B, we have some different force considerations. Assuming that car A and car B are complete mirrors of each other (again, this is a highly idealized situation), they would collide with each other going at precisely the same speed (but opposite directions).

From conservation of momentum, we know that they must both come to rest. The mass is the same. Therefore, the force experienced by car A and car B are identical and are identical to that acting on the car in case A.

This explains the force of the collision, but there is a second part of the question—the energy considerations of the collision.

Energy

Force is a vector quantity while kinetic energy is a scalar quantity, calculated with the formula K = 0.5mv2.

In each case, therefore, each car has kinetic energy K directly before the collision. At the end of the collision, both cars are at rest, and the total kinetic energy of the system is 0.

Since these are inelastic collisions, the kinetic energy is not conserved, but total energy is always conserved, so the kinetic energy "lost" in the collision has to convert into some other form - heat, sound, etc.

In case A, there is only one car moving, so the energy released during the collision is K. In case B, however, there are two cars moving, so the total energy released during the collision is 2K. So the crash in case B is clearly more energetic than the case A crash, which brings us to the next point.

From Cars to Particles

Why do physicists accelerate particles in a collider to study high-energy physics?

While glass bottles shatter into smaller shards when thrown at higher speeds, cars don't seem to shatter in that way. Which of these applies to atoms in a collider?

First, it's important to consider the major differences between the two situations. At the quantum level of particles, energy and matter can basically swap between states. The physics of a car collision will never, no matter how energetic, emit a completely new car.

The car would experience exactly the same force in both cases. The only force that acts on the car is the sudden deceleration from v to 0 velocity in a brief period of time, due to the collision with another object.

However, when viewing the total system, the collision in case B releases twice as much energy as the case A collision. It's louder, hotter, and likely messier.

In all likelihood, the cars have fused into each other, pieces flying off in random directions.

And this is why colliding two beams of particles are useful because in particle collisions you don't really care about the force of the particles (which you never even really measure), you care instead about the energy of the particles.

A particle accelerator speeds particles up but does so with a very real speed limitation (dictated by the speed of light barrier from Einstein's theory of relativity). To squeeze some extra energy out of the collisions, instead of colliding a beam of near-light speed particles with a stationary object, it's better to collide it with another beam of near-light speed particles going the opposite direction.

From the particle's standpoint, they don't so much "shatter more," but definitely when the two particles collide more energy is released. In collisions of particles, this energy can take the form of other particles, and the more energy you pull out of the collision, the more exotic the particles are.

Conclusion

The hypothetical passenger would not be able to tell any difference whether he was colliding with a static, unbreakable wall or with his exact mirror twin.

The particle accelerator beams get more energy out of the collision if the particles are going in opposite directions, but they get more energy out of the total system—each individual particle can only give up so much energy because it only contains so much energy.