A particle falls from a height h upon a fixed horizontal plane an… - Vidyadip

MCQ

A particle falls from a height $h $ upon a fixed horizontal plane and rebounds. If $e$ is the coefficient of restitution, the total distance travelled before rebounding has stopped is

✓
$h\left( {\frac{{1 + {e^2}}}{{1 - {e^2}}}} \right)$
B
$h\left( {\frac{{1 - {e^2}}}{{1 + {e^2}}}} \right)$
C
$\frac{h}{2}\left( {\frac{{1 - {e^2}}}{{1 + {e^2}}}} \right)$
D
$\frac{h}{2}\left( {\frac{{1 + {e^2}}}{{1 - {e^2}}}} \right)$

✓

Answer

Correct option: A.

$h\left( {\frac{{1 + {e^2}}}{{1 - {e^2}}}} \right)$

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)$

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