# Relative motion and collisions

1. Feb 4, 2012

### imcurious

My understanding of relativity is that if two masses are in constant motion relative to each other, either one of them could be considered at rest with the other moving and it makes no difference which. However thinking about a collision between a big truck and a small car makes me wonder about this. Suppose that there is a huge 18 wheeler truck barreling down the highway toward a Volkswagen stalled in the middle of the road. Then suppose that Samantha the witch blinks out the road and everything else in the universe except the truck and the Volkswagen. When the fast-moving truck hits the Volkswagen, it will surely go flying for a considerable distance. However if we reverse the situation, and have the truck stalled in the middle of the highway and the Volkswagen striking the truck (with them being alone in the universe), the collision would have quite a different outcome, with the car merely splatting against the truck and not flying anywhere. The point here is that it does make a difference which one is at rest and which one is in motion and one cannot just arbitrarily choose one to be at rest and the other in motion. Does anyone have any comments on this? I'm confused.

2. Feb 4, 2012

### Staff: Mentor

What you describe is correct. It has nothing to do specifically with special relativity. What you describe is correct in Galilean relativity also. The velocities before the collision are relative, so are the velocities after the collision.

Last edited: Feb 4, 2012
3. Feb 4, 2012

### Michael C

Careful: you just said "stalled in the middle of the highway". But you have already stated that the highway no longer exists: the only things in this universe are the truck and the VW. You can look at the collision from the point of view of the truck, or from the point of view of the VW, or indeed from any other point of view: it'll still be the same collision and nobody will be able to say that one of the vehicles was "at rest".

What all observers will agree on is that the change in velocity of the VW is much more than the change in velocity of the truck.

4. Feb 4, 2012

### ghwellsjr

When you do your experiment on a highway with gravity holding the vehicles to the pavement and offering tremendous friction, especially when the brakes are applied, then yes, there will be a huge difference in the outcome of the collision. But if you instead put the vehicles on a road covered with ice so that there is no appreciable friction and repeat the collision, what you will see is that "When the fast-moving truck hits the Volkswagen, it will surely go flying for a considerable distance", but so will the truck. It's speed will be hardly affected by the impact and the two will travel down the highway together.

If, on the other hand, the truck was stalled and the car hits it on the icy road, the truck will only move slightly and "the car merely splatting against the truck and not flying anywhere."

The point is that if there were another car traveling at constant velocity alongside the moving truck and videoing the collision it would capture the same image (ignoring the background) as it would in the second scenario when it and the truck are stopped on the ice and the moving car hits it. Don't you agree?

Or you could capture two videos from the point of view of the initial state of the car, first stopped on the icy highway which would show the truck and the car go flying out of view of the video, or second, with the video car traveling alongside the car as it impacts the truck but because the video car continues on without stopping, the image shows the collision with both vehicles going out of view of the video.

Does that make sense to you?

Last edited: Feb 4, 2012
5. Feb 4, 2012

### pervect

Staff Emeritus
This problem really doesn't have to do with relativity at all.

You write
In order to eliminate the road, you need to imagine either a frictionless surface, or a collision in space, and apply Newton's laws. Because if you are imagining a road with friction, the road is important to the problem.

And, you won't have either driver applying the brakes after the collision (another side effect of "no friction")

There two main limiting sorts of collisions one can consider in Newtonian mechanics (and there are the relativistic counterparts) elastic collisions and inelastic collisions.

IF you have an elastic collision, the small object hitting the more massive will dart away at exactly the same velocity that it impacted, with no energy loss.

More typical of the car collision is the inelastic collision. In this case, the small object and the large object will stick together, with zero relative velocity.

There are also intermediate cases - semi-elastic collisions.

Do you know how to properly analyze the collision of a large mass and a small mass in Netonian physics? If you do, I'd suggest that you carry out the mathematical analysis. If you don't, I'd suggest asking in another forum how to carry out the analysis, and understand first the Newtonian predictions for the various sorts of collision, then rethink the problem.

What you should find from a Newtonian analysis is that there will be some constant relative velocity between the two masses before the collision, and some other constant relative velocity after the two masses collide (in the absence of friction).

For an elastic collision the ratio of the pre-impact and post-impact speeds will be 1, for a perfectly inelastic collision the post-impact speed will be zero, and for a partially elastic colllision the ratio will be between 0 and 1.

6. Feb 6, 2012

### imcurious

In a perfectly elastic collision, yes I can see that the result would be the same regardless of which is considered to be at rest, but in a real workd oartially elastic collision like I described above, some energy of the collision will be absorbed by the car and the truck resulting in damage,. If the miore massive body (the truck) is moving, and the less massive body (the car) is at rest, the kinetic energy involved in the collision is greater than if the less massive body is moving, and the more massive body is at rest. Therefore the collision has a different result depending on which body is in motion. It seems that velocity rekative to some reference frame that contains both the car and the truck, must be considered rather than their velocity relative to each other.

7. Feb 6, 2012

### harrylin

More precisely (and that is essential here!), depending on which body is in motion relative to the road, if you want to account for friction with the road after the collision.
In your example perhaps the motions of three bodies play a role, and therefore all three of them must be considered. On the other hand, if you neglect damage to the road (or speed relative to the road) then you don't need to include it.
Note: the kinetic energy calculated relative to a reference frame has no direct bearing on the outcome of the collision. You can verify this yourself by doing an example calculation.

Last edited: Feb 6, 2012
8. Feb 6, 2012

### Staff: Mentor

This is true.

But this doesn't follow from the above.

Why don't you work out a concrete example involving a non-relativistic completely inelastic collision between a 1000 kg car moving 10 m/s and a stationary 10000 kg truck. Calculate the change in KE for each vehicle and the change in total KE. The change in total KE is the energy that goes into heat and mechanical deformation of the car and truck. Then, re-do the calculation in the frame where the car is at rest and the truck is moving at -10 m/s.

You will see how conservation of momentum and conservation of energy work together to avoid your "therefore ..." above.

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