A partical 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

Options

A. `h((1 + e^(2))/(1 - e^(2)))`
B. `h((1 - e^(2))/(1 + e^(2)))`
C. `(h)/(2) ((1 - e^(2))/(1 + e^(2)))`
D. `(h)/(2) ((1 + e^(2))/(1 - e^(2)))`

Answer

D

Explanation

Height attained by the ball after the first collision ,
`h_(1) = e^(2) h` , after the second collision `h_(2) = e^(4) h ,….`
Total distance traveled
`S = h + 2h_(1) + 2h_(2) + … oo`
` = h + 2e^(2) h( 1 + e^(2) + … oo)`
`= h + 2e^(2) h (1 + e^(2) + ... oo)`
`= h + 2 e^(2)h ((1)/(1 - e^(2))) = ((1 + e^(2))/(1 - e^(2)))h`
Recall : Sum of infinite terms in geometric progression `(GP)`
`a + ar + ar^(2) + ... oo`
`S_(oo) = (a)/( 1 -r) , r lt 1`