Two particles having masses m_1 and m_2 undergo a head-on...

Two particles having masses $m_1$ and $m_2$ undergo a head-on collision. Their initial speeds are $u_1$ and $u_2$, travelling in the same direction. The collision commences at $t = 0$ and the particles interact over a time interval $\Delta t$. Throughout the collision, the speed of the first particle changes according to the relation
$v(t) = u_1 + \dfrac{t}{\Delta t}(v_1 - u_1)$.
Determine the speed of the second particle as a function of time while the collision takes place.