Hailstorms are observed to stike the surface of a frozen lake at an angle of `30^(@)` with the vertical and rebound at an angle of `60^(@)` with vertical. Assuming the contact to be smooth, the coefficient of restitution is

Options

A. `1//4`
B. `1//3`
C. `1//5`
D. `3//4`

Explanation

In figure, we have shown initial velocity of hail storm `(` before impact `)= u ` at ` 30^(@)` with vertical. Its components are `u cos 30^(@)` along vertical and `u sin 30^(@)` along the horizontal.
After impact, velocity of hail storm is `upsilon` at `60^(@)` with vertical. Its components are `upsilon cos 60^(@)` alon the horizontal.
As the surface of frozen lake is smooth, the horizotal component remains unaffected.
`:. upsilon sin 60^(@)=u sin 30^(@) ...(i)`
The vertical component changes from `u cos 30^(@)` to `upsilon cos 60^(@)`. If `e` is coefficient of restitution, then
`u cos 60^(@)= e u cos 30^(@) ....(ii) ` Divid `(ii)` and `(i)`,
`(e u cos 30^(@))/( u sin 30^(@))=(upsilon cos 60^(@))/( upsilon sin 60^(@))`
or `e cot 30^(@)= cot 60^(@)`
`e=(cot 60^(@))/( cost 30^(@))=(1//sqrt(3))/(sqrt(3))=(1)/(3)`