a
(a)Particle falls from height h then formula for height covered by it in nth rebound is given by
\({h_n} = h{e^{2n}}\)
where \( e =\)  coefficient of restitution,\( n =\) No. of rebound
Total distance travelled by particle before rebounding has stopped
\(H = h + 2{h_1} + 2{h_2} + 2{h_3} + 2{h_n} + ........\)
\( = h + 2h{e^2} + 2h{e^4} + 2h{e^6} + 2h{e^8} + .........\)
\( = h + 2h({e^2} + {e^4} + {e^6} + {e^8} + .......)\)
\( = h + 2h\left[ {\frac{{{e^2}}}{{1 - {e^2}}}} \right] = h\,\left[ {1 + \frac{{2{e^2}}}{{1 - {e^2}}}} \right] = h\,\left( {\frac{{1 + {e^2}}}{{1 - {e^2}}}} \right)\)