Question

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)`.

Answer 1

(a) Displacement in `1 s` or `1^(st)` second
`s_(1)=ut+1/2at^(2)=0+1/2a(1)^(2)=1/2a`
or `s_(1)=u+1/2a(2t-1)=0+1/2a(2xx2-1)=1/2a`
Displacement in the `2^(nd)` second
`s_(2)=u+1/2a(2t-1)=0+1/2a(2xx2-1)=3/2a`
Displacement in the `3^(rd)` second
`s_(3)=0+1/2a(2xx3-1)=5/2a`
`s_(1):s_(2):s_(3): ...=1/2a:3/2a:5/2a: ...`
`=1:3:5: ...`
(b)
A to B: Displacement in the first `t_(0)` second
`s=ut+1/2at^(2)`
`s_(1)=0+1/2 at_(0)^(2)=1/2at_(0)^(2)`
`A` to `C`: `t=t_(0)+t_(0)=2t_(0)`
`s_(1)+s_(2)=0+1/2a(t_(0)+t_(0))^(2)=4.(1)/(2)at_(0)^(2)`
Displacement in the next `t_(0)` second
`s_(2)=3/2at_(0)^(2)`
`A` to `D`: `t=t_(0)+t_(0)+t_(0)=3t_(0)`
`s_(1)+s_(2)+s_(3)=0+1/2a(t_(0)+t_(0)+t_(0))^(2)=9(1)/(2)at_(0)^(2)`
Displacement in the next `t_(0)` second
`s_(3)=51/2at_(0)^(2)`
`s_(1):s_(2):s_(3)=1:3:5`