Extreme Momentum

At time $t=0$ , a stationary particle of mass $m$ starts to experience a time dependent force $F(t) = \begin{cases} \mathbf{k}t(T-t) & \text{for } 0 \leq t \leq T \\ 0 & \text{for } t \gt T \end{cases}$

where $\mathbf{k}$ is a constant vector.

Find the momentum of the particle when the action of the force discontinues.

• If the answer is in the form $\displaystyle \frac { \mathbf{k}{ T }^{ \alpha } }{ \beta }$ , where $\alpha$ and $\beta$ are natural numbers, find $\alpha + \beta$ .