A cylindrical hall has a horizontal smooth floor. A ball is projected along the floor from A point on the wall in a direction making an angle `theta` with the radius through the point the ball returns bak to the initial point after two impacts with the wall. if the coefficient of restitution is e than `tan^(2)theta` will be

Options

A. `(1+e+e^(2))/(e^(3))`
B. `(1+e)/(e^(2))`
C. `(e^(2))/(1+e)`
D. `(e^(3))/(1+e+e^(2))`

Answer

D

Explanation

`tanalpha=(usintheta)/(eucostheta)=(tantheta)/(e)`
`thereforealpha=(1)/(2)tantheta`
`thereforebeta=(1)/(e)tantheta`
`2theta+2alpha+2beta=180^(@)`
`alpha+beta=(90-theta)`
`tan(alpha+beta)=cottheta`
`implies(tanalpha+tanbeta)/(1-tanalphatanbeta)=(1)/(tantheta)`
`implies((1)/(e)tantheta+(1)/(e^(2))tantheta)/(1-(1)/(e^(3))tan^(2)theta)=(1)/(tantheta)`
`impliestan^(2)theta((1)/(e)+(1)/(e^(2)))=1-(1)/(e^(3))tan^(2)theta`
`impliestan^(2)theta((1)/(e)+(1)/(e^(2))+(1)/(e^(3)))=1`
`tan^(2)theta((1)/((1)/(e)+(1)/(e^(2))+(1)/(e^(3))))=(e^(3))/(1+e+e^(2))`