Successive direct impactsEdexcel A-Level Further Maths: Revision notes

In a successive impact problem the spheres collide more than once, or a sphere hits a smooth wall or floor between collisions. Treat every impact separately: the velocities just after one impact are the velocities just before the next. For each impact use conservation of momentum (sphere-sphere) or $v=eu$ (sphere-wall), together with Newton's law of restitution. Keep one positive direction throughout.

A ball hits a smooth floor at speed $u$ and rebounds at speed $eu$. The speed is multiplied by $e$ at every impact: after $n$ impacts it is $e^nu$. Under gravity the ball returns to the floor at the speed with which it left, and the height reached depends on speed squared, so heights reduce by a factor of $e^2$ each bounce. Example: $e=0.5$, dropped from $3.2$ m. The first rebound reaches $0.25\times3.2=0.8$ m and the second $0.2$ m.

When $A$ hits $B$, and then $B$ hits $C$, solve the $A$–$B$ collision first. The velocity of $B$ after that impact is the velocity with which $B$ hits $C$. For equal masses with the second sphere at rest, $v_1=\frac{u(1-e)}{2}$ and $v_2=\frac{u(1+e)}{2}$. Example: $u=6$ m s$^{-1}$, $e=\frac12$ gives $1.5$ and $4.5$ m s$^{-1}$.

If a sphere rebounds from a wall it can hit another sphere again. Work out: the velocities after the first sphere-sphere impact, the rebound from the wall ($eu$ for the wall's own $e$), and then the next collision with the signed velocities. Continuing the example, if the wall's coefficient is $\frac13$ the second sphere rebounds at $\frac13\times4.5=1.5$ m s$^{-1}$ away from the wall, towards the first sphere, which is still moving towards the wall at $1.5$ m s$^{-1}$. They approach, so they collide again.

Another impact occurs if two spheres are moving towards each other, or the rear sphere is moving faster than the front sphere in the same direction, or a sphere is moving towards a wall. If the speeds are in order (rear sphere not faster than the one ahead, and none heading for a wall) there are no more impacts. To show a further collision, state the two velocities and the reason: for example, '$A$ moves at $3$ m s$^{-1}$, faster than $B$ at $1.5$ m s$^{-1}$, so $A$ catches $B$'.