A particle falls from a height h on a fixed horizontal surface an… - Vidyadip

MCQ

A particle falls from a height h on a fixed horizontal surface and rebounds. If e is the coefficient of restitution, then the total distance travelled by the particle before it stops rebounding is

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

✓

Answer

Correct option: A.

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

(A)

Total distance $=h+2 e^2 h+2 e^4 h \ldots$.
$=h+2 e^2 h\left(1+e^2+\ldots\right)$
Using binomial expansion,
$\left(1+ e ^2+ e ^4+\ldots\right)=\frac{1}{\left(1- e ^2\right)}$
$= h +\frac{2 e ^2 h}{1- e ^2}$
$=\frac{ h - e ^2 h+2 e ^2 h}{\left(1- e ^2\right)}$
$=\frac{ h \left(1+ e ^2\right)}{\left(1- e ^2\right)}$

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