# Relative velocity of two cars crashing into each other

• ninuss
In summary, two cars of equal mass crashing into each other with the same relative velocities will produce the same kind of damage. However, the energy available to damage the cars is not the total kinetic energy, but rather the difference between the initial and final kinetic energies. This means that the second crash, where both cars are moving at the same speed in the same direction, may have a different practical effect due to the speed of each car relative to the road and air. Further investigation and calculations would be needed to determine the exact differences in this scenario.
ninuss

And gathered what I already thought was correct, that the relative velocities of two cars of equal mass crashing into each other are the same in examples 1) and 2),

where in example 1), car X is moving a 100km/h in a straight line towards car Y, and car Y is moving at 100km/h in a straight line towards car X;

and in example 2), car X is moving at 200km/h in a straight line towards car Y, and car Y is moving at 0km/h towards car X (car Y is standing still)

Now, my question is, would the crashes in these two examples produce the same kind of damage to the cars? I was told that in example 2, the kinetic energy released is twice as much as in example 1, because kinetic energy is calculated by squaring velocity, and so simply adding the velocity of each car and then squaring it, is not the same as squaring each and then adding it.

But my thinking is that if the relative velocity is the same, the crash should be the same..

In example 1), both cars would roughly come to a full stop (or they may bounce a little), and in example 2), after the crash, both cars would continue moving in the same direction as car X, but at a slower speed. That's the only difference I can come up with. But it still makes no sense to me that the damage would be doubled when the relative velocity is the same.

Thing to do is work it out in terms of momentum (which is conserved) and check the kinetic energy loss (kinetic energy is not conserved: the remainder goes into crushing the cars).

for simplicity use the same mass for each car

My bet is you can prove yourself correct !

Note that the energy available to damage the cars is not the total kinetic energy. It's the difference between the initial and final kinetic energies. As BvU says, you can calculate this and the result should be the same in both cases.

Note also that this is only the effect of the initial collision. What happens next does depend on speed relative to the Earth - there is a big difference between stationary wreckage and wreckage doing 100km/h.

PeroK
ninuss said:
Now, my question is, would the crashes in these two examples produce the same kind of damage to the cars? I was told that in example 2, the kinetic energy released is twice as much as in example 1, because kinetic energy is calculated by squaring velocity, and so simply adding the velocity of each car and then squaring it, is not the same as squaring each and then adding it.

But my thinking is that if the relative velocity is the same, the crash should be the same..

In example 1), both cars would roughly come to a full stop (or they may bounce a little), and in example 2), after the crash, both cars would continue moving in the same direction as car X, but at a slower speed. That's the only difference I can come up with. But it still makes no sense to me that the damage would be doubled when the relative velocity is the same.

From the basic point of view, the two crashes are the same. Kinetic energy is frame dependent, so you could study the second crash from a car traveling at ##100 km/h## and, in this frame, kinetic energy of the two cars is the same.

But, there is a critical practical difference between the two crashes: the speed of each car relative to the road and the air. It would be interesting to know what practical differences this might make to the scenario, from an accident investigation point of view!

BvU said:
My bet is you can prove yourself correct !

I hope so! :)

Ibix said:
Note that the energy available to damage the cars is not the total kinetic energy. It's the difference between the initial and final kinetic energies. As BvU says, you can calculate this and the result should be the same in both cases.

PeroK said:
From the basic point of view, the two crashes are the same. Kinetic energy is frame dependent, so you could study the second crash from a car traveling at ##100 km/h## and, in this frame, kinetic energy of the two cars is the same.

Awesome! Thanks a lot!

Ibix said:
Note also that this is only the effect of the initial collision. What happens next does depend on speed relative to the Earth - there is a big difference between stationary wreckage and wreckage doing 100km/h.

PeroK said:
But, there is a critical practical difference between the two crashes: the speed of each car relative to the road and the air. It would be interesting to know what practical differences this might make to the scenario, from an accident investigation point of view!

Very interesting points!

## 1. How is the relative velocity of two cars calculated in a crash?

The relative velocity of two cars in a crash is calculated by subtracting the velocity of one car from the velocity of the other. This gives the net velocity at which the cars are approaching each other.

## 2. Does the mass of the cars affect the relative velocity in a crash?

Yes, the mass of the cars does affect the relative velocity in a crash. The heavier the cars, the more force is needed to change their velocity, resulting in a higher relative velocity at impact.

## 3. Can the relative velocity of two cars in a crash be negative?

Yes, the relative velocity of two cars in a crash can be negative. This occurs when the cars are moving in opposite directions and have different velocities, resulting in a negative net velocity.

## 4. What role does momentum play in determining the relative velocity of two cars in a crash?

Momentum plays a crucial role in determining the relative velocity of two cars in a crash. The total momentum of the two cars before the crash is equal to the total momentum after the crash, which helps to determine the relative velocity at impact.

## 5. How does the coefficient of restitution affect the relative velocity of two cars in a crash?

The coefficient of restitution, which represents the elasticity of the materials in the cars, can affect the relative velocity in a crash. A higher coefficient of restitution means that the cars will bounce off each other with more force, resulting in a higher relative velocity at impact.

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