I came across a problem in physics and I'm struggling a bit with getting the right intuition behind it. Below is the problem, my answer attempt and my question.
Problem: Two small spheres $P$ and $Q$ of equal mass lie touching each other in a smooth circular horizontal groove of radius $a$. Sphere $P$ is projected away from $Q$ with speed $U$. Assuming no resistance to motion in the groove (so that between collisions the speeds of the balls are constant), and a coefficient of restitution $e$ of each collision, find the time that elapses before the $n$th collision.
Assume that the speeds of $P$ and $Q$ after the $n$th collision satisfy
$$
V_n^P-V_n^Q=(-e)^nU,\\
V_n^Q+V_n^P=U.
$$
My attempt: The time $\Delta t_1$ elapsed before the first collision is given by $\Delta t_1=2\pi a/U$. For the time between first and second collision, $\Delta t_2$, we first notice that, for some $x$,
$$
V_1^P=\frac{x}{\Delta t_2}\,\,\text{ and }\,\,V_1^Q=\frac{2\pi a+x}{\Delta t_2}.
$$
Thus
$$
\Delta t_2=\left|\frac{2\pi a}{V_1^Q-V_1^P}\right|=\frac{2\pi a}{eU}.
$$
The same argument could be used to calculate the remaining terms up to $\Delta t_{n}$. Then, simply sum these terms.
Question: My answer is done assuming that the movement of the two spheres is always kept in the same direction after a collision, thus the $2\pi a +x$. If the spheres were to bounce in opposite directions after every collision, I'd need to to use $2\pi a-x$ instead, which would yield a different result. How can I distiguinsh the two cases, and how would I know they would keep the same movement direction from the way the question is formulated?
Any insight is appreciated.
Best Answer
In the meantime, I think I've solved it. Here's my solution.
The reason both spheres move in the same direction (clockwise, for example) after a collision is due to the fact that the collision is inelastic ($e<1$). If it was perfectly elastic ($e=1$), sphere $P$ would stop and only $Q$ would move after the first collision (as seen in Newton's cradle). To see this, consider simply the case where two spheres, $1$ and $2$, of equal mass collide. Sphere $2$ is initially at rest ($u_2=0$), and sphere $1$ has initial velocity $u_1$. Then, conservation of momentum and restitution yields \begin{align*} \begin{cases} u_1=v_1+v_2\\ v_2-v_1=eu_1 \end{cases} \end{align*} Thus, solving for $v_1$, $$ v_1=u_1-v_2=u_1-eu_1-v_1 \Rightarrow v_1=(1-e)u_1/2. $$ Notice that $u_1,u_2,v_1$ and $v_2$ are velocities, i.e, they are vectors. Hence, if $e=1$, we get $v_1=0$ and if $e<1$, since $1-e>0$, we have that $v_1$ has the same direction as $u_1$ and $v_2$.