A particle is dropped from certain height. The time taken by it to fall through successive distance of 1 m each will be

Options

A. all equal, being equal 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., `sqrt1,(sqrt2-sqrt1),(sqrt3-sqrt2),(sqrt4-sqrt3),.....`
D. in the ratio of the reciprocals of the square roots of the integers, i.e., `1/sqrt1,1/sqrt2,1/sqrt3,........`

Answer

C

Explanation

Time taken to cover n meters is given by
`n=1/2"gt"_n^2 " or " t_(n) =sqrt((2n)/g)" "( :.u=0)`
Time taken to cover (n + 1) metres is given by
`t_(n+1)=sqrt((2(n+1))/g)`
`:. t_(n+1) -t_n=sqrt((2(n+1))/g)-sqrt((2n)/g)=sqrt(2/g)[sqrt(n+1)-sqrtn]`
This gives ratio as :
`sqrt1, (sqrt2-sqrt1),(sqrt3-sqrt2),.....,` etc.