A particle starts moving rectilinearly at time `t=0` such that its velocity `v` changes with time `t` according to the equation `v=t^(2)-t`, where `t` is in seconds and `v` in `ms^(-1)`. Find the time interval for which the particle retards.

Options

A. `(1)/(2) lt t lt 1`
B. `(1)/(2) gt t gt 1`
C. `(1)/(4) lt t lt 1`
D. `(1)/(2) lt t lt (3)/(4)`

Answer

A

Explanation

Acceleration of the particle `a= 2t-1` The particle retards when acceleration is opposite to velocity.
`rArr a, v lt 0 rArr (2t -1) (t^(2)-1) lt 0 rArr t (2t-1) (t-1) lt 0`
Now t is always positive
`:. (2t-1) (t-1) lt 0`
or `2t-1 lt 0 and t-1 gt 0 rArr t lt (1)/(2) and t gt 1` This is not possible or `2t-1 gt 0 "&" t-1 lt 0 rArr 1//2 lt t lt 1`