From a tower of height `H`, a particle is thrown vertically upwards with a speed `u` . The time taken by the particle , to hit the ground , is ` n` times that taken by it to reach the highest point of its path. The relative between ` H, u and n` is :

Options

A. `2gH = n u^2 (n - 2)`
B. `gH = (n - 2) u^2`
C. `2gH = n^2 u^2`
D. `gH = (n - 2)2 u^2`

Answer

A

Explanation

When a particle is thrown vertcally upwads with a speed `u`, at highest point the velocity of the particle will be zero.
Using `v = u + at`
`0 = u - g t rArr t = (u)/(g)` …(i)
The time taken by the particle, to hit the ground, is `n` times that taken by it to reach the highest point of its path Let `t_1 = nt`
Using, `s = ut_1 + (1)/(2) at_1^2`
`rArr -H = u(nt) - (1)/(2) g(nt)^2`
`rArr -H = u xx n((u)/(g)) - (1)/(2) gn^2 ((u)/(g))^2` [using (i)]
`rArr 2gH = 2n u^2 - n^2 u^2 rArr 2gH = n u^2 (n - 2)`.