Momentum-Impulse Theorem

Momentum-Impulse Theorem

The change in the momentum of a point mass between the moments $t_1$ and $t_2$ is equal to the impulse of the net external force acting on it during that rime interval.

$$\mathbf{J}=\mathbf{p}(t_2)-\mathbf{p}(t_1)=\Delta \mathbf{p}$$

PROOF

According to Newton’s second law of motion, the net external force $\mathbf{F}$ is the the derivative of momentum.

$$\mathbf{F}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}$$

We just then substitute in the definition for impulse:

$$\mathbf{J}={\int }_{t_1}^{t_2}\mathbf{F}\mathrm{dt}={\int }_{t_1}^{t_2}\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}\mathrm{dt}=\mathbf{p}(t_2)-\mathbf{p}(t_1)=\Delta \mathbf{p}$$