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.

Explanation

Velocity `v=t^(2)-t=t(t-1)`
Acceleration, `a=(dv)/(Dt)=2t-1`
Motion of particle is considered as retarding motion when v and a have opposite signs that means they are along opposite directions.
From equation 1 we can see that,
velocity is positive when `t gt 1`
velocity is negative when `t lt 1`
From equation 2 we can see that,
Acceleration is positive when `t gt 1//2`
Acceleration is negative when `t lt 1//2`
From above results we can see that when velocity is positive then acceleration cannot be negative
But there is an interval `(1)/(2) lt t lt1`, when velocity is negative and acceleration is positive, hence this is the interval of time when particle retards.