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I'm confused about collisions where there are parts of the objects not involved in it. How do these parts move after the collision? I'll make an example.

Consider a cart with an incline attached (total mass of the two $m_A$) to it and a cylinder (mass $m$) which is free to move with no friction on the cart. Initially the system is moving at velocity $v_0$. Then the cart collides with a second cart (mass $m_B$), initially steady, and the two remain attached. Find the height on the incline reached by the cylinder.

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I was quite sure that in such situation the cylinder just continues its motion with velocity $v_0$ but if I try to solve the problem in this way I get the wrong answer.

The final velocity of the two carts is $v_f=\frac{m_A v_0}{m_A+m_B}$

Then I use conservation of energy for the cylinder (in relative motion on the cart) $\frac{1}{2} m (v_0-v_f)^2=m g h$

While the answer is

$h=\frac{m_B^2 v_0^2}{2(m_A+m_B)(m_A+m_B+m)g}$

What is wrong in thinking that the cylinder will continue its motion at constant velocity? It is not involved in the collision (there is not even friction on the cart).

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closed as off-topic by CuriousOne, John Rennie, ACuriousMind, user36790, Qmechanic Mar 22 '16 at 12:40

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Well, the trick is to think about the situation when the cylinder has reached its highest point: how is it moving relative to the carts then, and what does conservation of momentum momentum say compared with the situation just after the collision?

(Note I am intentionally not giving you the complete answer but what is, I hope, an adequate hint, as I don't want to just answer a homework problem but I think this is more interesting than that, as there is quite a subtle point here.)

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Your approach is almost correct except for the fact that as the cylinder is climbing the incline, it is pushing against the incline so incline plus carts one and two accelerate while the cylinder decelerates.

The way to tackle this is to solve the problem in the center of mass frame - the carts do not maintain a constant speed while the cylinder rolls up the incline, but the center of mass does.

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