A particle is dropped from rest from a large height Assume `g` to be constant throughout the motion. The time taken by it to fall through successive distance of `1 m` each will be :

Options

A. All equal, being to `sqrt(2//g)` second
B. In the ratio of the square roots of the integers 1,2,3,…
C. In the ratio of the difference in the square roots of the integers, i.e., `sqrt(1), (sqrt(2)-sqrt(1))(sqrt(3)-sqrt(2)),(sqrt(4)-sqrt(3))`…
D. In the retio of the reciprocal of the square roots of the integers, i.e., `1/sqrt(1), 1/sqrt(2),1/sqrt(3),1/sqrt(4)`

Answer

C

Explanation

`1=1/2g t_(1)^(2)impliest_(1)=sqrt(2/g)`
`1+1=1/2g(t_(1)+t_(2))^(2)`
`impliest_(1)+t_(2)=sqrt(4/g)`
`1+1+1=1/2g(t_(1)+t_(2)+t_(3))^(2)`
`impliest_(1)+t_(2)+t_(3)=sqrt(6/g)`
`t_(2)=sqrt(4/g)-sqrt(2/g)`
`t_(3)=sqrt(6/g)-sqrt(4/g)`
`t_(1):t_(2):t_(3): ...=sqrt(2/g):sqrt(4/g)-sqrt(2/g):sqrt(6/g)-sqrt(4/g):`
`=sqrt(1):(sqrt(2)-sqrt(1)):(sqrt(3)-sqrt(2)):`