Principle of impulse and momentum

Principle

Linear impulse of a force is $\int \mathbf{F}\,dt$ (unit N·s). Linear momentum is $m\mathbf{v}$ (unit kg·m/s). The principle states that the linear impulse of the resultant force acting on a particle during a time interval equals the change in its linear momentum:

$$m\mathbf{v}_1 + \int_{t_1}^{t_2} \mathbf{F}\,dt = m\mathbf{v}_2$$

Derivation

Newton's second law: $\mathbf{F} = m\dfrac{d\mathbf{v}}{dt}$ (constant mass). Multiply by $dt$ and integrate from $t_1$ to $t_2$:

$$\int_{t_1}^{t_2}\mathbf{F}\,dt = m\int_{\mathbf{v}_1}^{\mathbf{v}_2} d\mathbf{v} = m\mathbf{v}_2 - m\mathbf{v}_1$$

It is a vector equation, so it is applied separately in each direction. It is useful when forces depend on time and when time (not distance) is required.

Impulsive force

An impulsive force is a very large force acting for a very short time (for example in a collision or a hammer blow), so that its impulse is finite while the displacement during the action is negligible. Non-impulsive forces such as weight and spring forces are ignored during the impact.

Conservation of linear momentum

For a system of particles, the internal forces occur in equal and opposite pairs (Newton's third law), so their impulses cancel. If the external impulse in a direction is zero, the total momentum of the system in that direction is the same before and after:

$$\sum m_i \mathbf{u}_i = \sum m_i \mathbf{v}_i$$

Examples are an explosion, a recoiling gun, and a collision between two bodies.