A particle starts from rest and moves under constant acceleration in a straight line. Find the ratio of displacement (a) in successive second and (b) in successive time interval `t_(0)`.

Explanation

(i) Displacement in 1 s or 1st second
`s_(1)=ut+(1)/(2)at^(2)=0+(1)/(2)a(1)^(2)=(1)/(2)a`
or `s_(1)=u+(1)/(2)a(2t-1)=0+(1)/(2)a(2xx1-1)=(1)/(2)a`
Displacement in the 2nd second
`s_(2)=u+(1)/(2)a(2t-1)=0+(1)/(2)a(2xx2-1)=(3)/(2)a`
Displacement in the 3rd second
`s_(3)=0+(1)/(2)a(2xx3-1)=(5)/(3)a`
`s_(1):s_(2):s_(3): ...=(1)/(2)a:(3)/(2)a:(5)/(2)a: ...=1:3:5: ....`

A to B : Displacement in the first `t_(0)` second
`s=ut+(1)/(2)at^(2)rArr s_(1)=0+(1)/(2)at_(0)^(2)=(1)/(2)at_(0)^(2)`
A to C `t=t_(0)+t_(0)=2t_(0)`
`s_(1)+s_(2)=0+(1)/(2)a(t_(0)+t_(0))^(2)=4(1)/(2)at_(0)^(2)`
Displacement in the next `t_(0)` second
`s_(2)=(3)/(2)at_(0)^(2)`
A to D `t=t_(0)+t_(0)+t_(0)=3t_(0)`
`s_(1)+s_(2)+s_(3)=0+(1)/(2)a(t_(0)+t_(0)+t_(0))^(2)=(9)/(2)at_(0)^(2)`
Displacement in the next `t_(0)` second
`s_(3)=(5)/(2)at_(0)^(2) rArr s_(1):s_(2):s_(3)=1:3:5`