Question

Two smooth spheres A and B with respective masses m and 2m and of equal radii rest on a smooth horizontal plane. A is projected towards B with speed u to collide direction with B. Given that the coefficient of restitution between A and B is e and that the speed of B after this is 4u/9, find:
(a) the value of e and the velocity of A after impact
(b) the magnitude of the impulse exerted by A on B.
(c) the percentage loss in kinetic energy during this impact.

Answer 1

a)

$e=\frac{\frac{4u}{9}}{u+\frac 13 u}=\frac 13,$

$v=u-(1+\frac 13)\frac{2m}{m+2m}(u-0)=\frac{1}{9}u,$

b)

$p=(1+\frac 13)\frac{m\cdot 2m}{m+2m}(u-0)=\frac 89 mu,$

c)

$T=(1-(\frac 13)^2)\frac{m\cdot 2m}{m+2m}\frac{(u-0)^2}{2}=\frac{8}{27}mu^2.$