A particle starts from rest and moves under constant acceleration in a straight line. Find the ratio of time in successive displacement `d`.

Explanation

`A` to `B`: `s=ut+1/2at^(2)`
`d=1/2at_(1)^(2)implies t_(1)=sqrt((2d)/a)`
`A` to `C`: `d+d=1/2a(t_(1)+t_(2))^(2)impliest_(1)+t_(2)=sqrt((4d)/a)`
`t_(2)=sqrt((4d)/a)-sqrt((2d)/a)`
`A` to `D`: `d+d+d=1/2a(t_(1)+t_(2)+t_(3))^(2)`
`implies t_(1)+t_(2)+t_(3)=sqrt((6d)/a)`
`t_(3)=sqrt((6d)/a)-sqrt((4d)/a)`
`t_(1):t_(2):t_(3):=sqrt((2d)/a):(sqrt((4d)/a)-sqrt((2d)/a)): (sqrt((6d)/a)-sqrt((4d)/a)) ...`
`=1:(sqrt(2)-1):(sqrt(3)-sqrt(2)): ...`