
(1) Let the object move at v_i in the positive x direction. Find the velocity of the object related to the floor after colliding the inside box.
(2) What is the position of the middle of the box after the second collision in the other side?
Answer:
(1) There is no external force; therefore, we can consider conservation of momentum. Let v and u be the velocities of the object and the box, respectively. The subscripts, i and f, represent initial and final. Thus, we have
\begin{equation} mv_i = mv_f + mu_f \end{equation}
The coefficient of restitution is defined as the relative velocity after collision divided by the relative velocity before collision. Namely,
\begin{equation} e = \frac{v_f - u_f}{0 - v_i} \end{equation}
From (1) and (2), we have
\begin{eqnarray} v_i &=& v_f + u_f \\ -v_i e &=& v_f - u_f \end{eqnarray}
Let's find v_f in terms of v_i by eliminating u_f.
\begin{equation} v_f = \frac{1-e}{2}v_i \end{equation}
(2) First, we find the final velocity of the box by eliminating v_f from (3) and (4).
\begin{equation} u_f = \frac{1+e}{2}v_i \end{equation}
The time from the first to the second collision is the distance, L, divided by the relative velocities.
\begin{equation} t = \frac{L}{u_f - v_f} = \frac{L}{ev_i} \end{equation}
Therefore, the distance that the box moved is
d = t \times u_f = \frac{L}{ev_i} \times \frac{1+e}{2}v_i = \frac{1+e}{2e}L
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