Relativity & Conservation of Momentum: A vs B Collision

[hprog]

Suppose A and B are in uniform motion toward each other, coming from the right and left respectively.
A claims B to move 100 mph to the left and B claims that A moves 100 mph to the right.
Now let us assume that A and B collide together and they crumple up into a combined object C, and let assume that friction and other factors were low, then according to the conservation of momentum the velocity has now to be the total of the velocities.
Will C now move to the left or the right?
If A was at rest, then the total velocity will be 100 to the left, and if B was at rest then the total should be 100 to the right.
So what is wrong here?

 

[PeterDonis]

Conservation of momentum doesn't say that "the velocity has to be the total of the velocities". Momentum is mass times velocity. So first of all we need to know the masses of A and B.

Let's assume that those masses are equal (which is probably what you meant). Let's also assume that the velocities are small compared to the speed of light, so we don't have to worry about any relativistic complications. Then we can write:

$$mv_{A} + mv_{B} = 2mv_{C}$$

where m is the common mass of A and B (so C, which is the two put together, has mass 2m). Note that each of the v's is a vector, so we have to know, not just their magnitudes, but their directions in order to evaluate the above equation. But, as you noted, the directions depend on whose reference frame we use. In A's frame, we have $v_{A} = 0$ and $v_{B} = -100$, whereas in B's frame, we have $v_{A} = 100$ and $v_{B} = 0$ (we're assuming that positive velocities are to the right and negative velocities are to the left). Plugging those numbers into the above equation will give you what $v_{C}$ must be in each frame.

It is true that the two answers, the one for A's frame and the one for B's frame, will be different. Is that what's bothering you?

 

[Dale]

So what is wrong here?

You need to distinguish between conservation and frame-invariance. Momentum is a conserved quantity, but it is not frame-invariant.

A frame-invariant quantity is one which is the same in all reference frames. Different frames disagree on v so they disagree on momentum. Momentum is not frame-invariant, or in other words momentum is relative to the reference frame.

A conserved quantity is one which does not change as a function of time in a given reference frame. Momentum is a conserved quantity. In other words, in a given reference frame its value before a collision is the same as its value after a collision.

 

[Rlam90]

The way in which the energy is transferred would depend on the makeup of the objects, would it not? If the objects were completely rigid and hit each other at an angle exactly parallel to their trajectory, they would bounce back in exactly opposite trajectories, correct? If the objects were made of smaller particles, the objects would react in accordance to their constituent particles and their structure withing the objects. My guess is that the objects would either crumple and bounce, shatter and bounce, or react and emit radiation. But I'm not a physicist.