Galilean Relativity & Kinetic Energy

[Raungar]

How would one find the reference frame in which an object [which undergoes a perfectly inelastic collision with a second object] experiences no change in its kinetic energy?

 

[A.T.]

How would one find the reference frame in which an object [which undergoes a perfectly inelastic collision with a second object] experiences no change in its kinetic energy?

Halfway (or rather half-speed) between initial and final rest frame (which in this case is the common center of mass frame).

 

[Raungar]

Alright, but suppose the first object starts at a constant velocity and the second at rest; without an initial frame at which both are at rest, the conjecture you present ceases to be useful. So, how might your answer adapt to fit the aforementioned situation?

 

[Orodruin]

ATs answer already includes this situation. The center of mass system is the system in which the total momentum (the sum of the momenta of both objects) is zero. There is no need for a system where both are at rest to define this frame.

Edit: However, it should be pointed out that while this frame satisfies your condition, it is not unique.

 

[Dale]

Alright, but suppose the first object starts at a constant velocity and the second at rest; without an initial frame at which both are at rest, the conjecture you present ceases to be useful. So, how might your answer adapt to fit the aforementioned situation?

A.T. is exactly correct. Your "suppose..." scenario is not relevant for this problem. You have to consider all reference frames and select the one which fits the problem's requirements. A.T. told you how to select it.

 

[Orodruin]

Some confusion might have arisen due to AT and us reading elastic instead of inelastic. AT's more general statement still holds: it is the system with the mean velocity of the objects initial and final velocities.

 

[Dale]

Some confusion might have arisen due to AT and us reading elastic instead of inelastic.

Oops, you are right. Usually this question is asked about an elastic collision.

The correct frame is indeed the one halfway between the initial and final rest frames (i.e. the object has equal velocity at the beginning and at the end), but it will most definitely not be the center of mass frame for an inelastic collision.

 

[Orodruin]

Well, to be more general, that frame is one such frame. Since there is only one condition and boosts form a 3d vector space (assuming 3d motion is allowed), more options exist.

 

[A.T.]

Some confusion might have arisen due to AT and us reading elastic instead of inelastic.

No, I read perfectly inelastic.

but it will most definitely not be the center of mass frame for an inelastic collision.

That part refereed the final rest frame, not the frame of constant KE. I now see it was ambiguously formulated. It should better read:

Average velocity of initial and final rest frame (the latter being the common center of mass rest frame in this case).

 

[A.T.]

assuming 3d motion is allowed, more options exist.

The OP doesn't explicitly state it must be an inertial frame, so there definitely are more options even in 1d, like the trivial rest frame of the object throughout the collision.

 

[Orodruin]

No, I read perfectly inelastic.

That part refereed the final rest frame, not the frame of constant KE. I now see it was ambiguously formulated. It should better read:

Average velocity of initial and final rest frame (the latter being the common center of mass rest frame in this case).

That indeed clears things up. I think everyone was reading it the other way and taking it from there.

 

[Dale]

That part refereed the final rest frame, not the frame of constant KE.

Wow! I am on a roll here. First I misunderstood the OP and then twice I misunderstood your post. I think I will just quit while I am behind!

 

[Raungar]

Thank you all for your input.