Question

A particle starts moving rectilinearly at j 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 $m{s}^{-1}$. f Find the time interval for which the particle retards.

Answer 1

Acceleration of the particle, $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=2t-1$
The particle retards when acceleration is opposite to velocity. Hence, the acceleration vector and velocity vector should be opposite to each other or the dot product of $\overrightarrow{a}$ and $\overrightarrow{v}$ should be negative.
$\Rightarrow$ $\overrightarrow{a}.\overrightarrow{v}<0$
$\Rightarrow$ $(2t-1)(t^2-t)<0$
t is always positive.
$\therefore$ (2t - 1)(t - 1) < 0
$\Rightarrow$ Either 2t - 1 < 0 or t - 1 > 0
$\Rightarrow$ t <$\frac{1}{2}$s and t > 1s. This is not possible.$\Rightarrow$ 2t - 1 >0 and t - 1 < 0 $\Rightarrow$ t >$\frac{1}{2}$s and t < 1s.Hence, the required time interval is $\frac{1}{2}s<t<1s$.