A body of mass m is moving in a straight line with momentum p ....

Given, $F = at$

when, $t = 0$, then linear momentum $= p$

when, $t = T$,~then linear momentum $= 2p$

According to Newton's law of motion, applied force, $F = \dfrac{dp}{dt}$

or~$dp = Fdt$

$dp = atdt$

or~${∫}_p^{2p}dp = a{∫}_{0}^{T}tdt$

\left(2p - p\right) = a \cdot \dfrac{T2}{2}$</p><p>$⇒ \dfrac{2p}{a} = T^{2} ⇒ T = \sqrt{\dfrac{2p}{a}}$