Linear Momentum of a Particle. Impulse of a Force

Linear momentum of a particle is a vector measure of mechanical motion defined as the product of particle mass and velocity.

Linear momentum

For a particle of constant mass:

$$\vec p=m\vec v.$$

The vector $\vec p$ has the same direction as velocity. Its SI unit is kg·m/s.

Impulse of a force

The impulse of a force from $t_1$ to $t_2$ is:

$$\vec J=\int_{t_1}^{t_2}\vec F\,dt.$$

For a constant force, $\vec J=\vec F\Delta t$. Impulse measures the action of a force over a time interval.

Impulse-momentum theorem

Newton's second law for constant mass gives $d\vec p/dt=\sum\vec F$. Integrating:

$$\vec p_2-\vec p_1=\int_{t_1}^{t_2}\sum\vec F\,dt.$$

Thus, the change in linear momentum equals the impulse of the resultant force.

Coordinate components

The vector theorem may be applied separately along coordinate directions, for example $mv_{2x}-mv_{1x}=\int\sum F_xdt$.

When the method is useful

The impulse approach is especially effective when velocities at two instants are needed but the detailed motion between them is not. It is also useful for large forces acting over short intervals.

Example

A 2 kg body initially moves at 3 m/s along the $x$ axis. A constant 8 N force acts in the same direction for 0.5 s. Its impulse is 4 N·s, so $2v_2-2\cdot3=4$ and $v_2=5$ m/s.

Common mistakes

• confusing linear momentum $m\vec v$ with kinetic energy;
• ignoring the vector nature of impulse;
• using $F\Delta t$ for a variable force without justification;
• omitting forces from the total impulse.