How Are Impulse and Momentum Related?

Impulse and momentum are linked by a simple rule: the net impulse on an object changes its momentum. This lets you analyze a short collision or impact without tracking every detail of the object’s motion during contact.

The same rule also explains momentum conservation in collisions. An individual object may gain or lose momentum, while the total momentum of a suitably chosen system remains unchanged if external forces deliver negligible impulse.

Momentum and impulse

The linear momentum of an object is its mass multiplied by its velocity:

$$\mathbf{p}=m\mathbf{v}.$$

Momentum is a vector: it points in the direction of the velocity. Its SI unit is kilogram-metre per second, equivalent to a newton-second.

The impulse delivered to an object over a time interval is the time integral of the net force on it:

$$\mathbf{J}=\int_{t_i}^{t_f}\mathbf{F}_{\text{net}}(t)\,dt.$$

Impulse is also a vector. If the net force is constant, this reduces to $\mathbf{J}=\mathbf{F}_{\text{net}}\Delta t$. If force varies, the impulse is the signed area under the force-versus-time graph, with each component considered separately.

The impulse–momentum theorem

Newton’s second law in momentum form is $\mathbf{F}_{\text{net}}=d\mathbf{p}/dt$. Integrating over the interval gives

$$\mathbf{J}=\Delta\mathbf{p}=\mathbf{p}_f-\mathbf{p}_i.$$

For an object whose mass does not change, this becomes

$$\mathbf{J}=m\mathbf{v}_f-m\mathbf{v}_i.$$

The theorem applies whether the force is steady or changes over time. It is especially useful for collisions, where the contact force may be large and difficult to model but acts for a short time. If you know the impulse and initial momentum, add the impulse vector to get the final momentum.

Worked example: a ball changing direction

A $0.15\,\text{kg}$ ball moves to the right at $20\,\text{m/s}$, then rebounds to the left at $15\,\text{m/s}$. Take rightward as positive. Its initial and final momenta are

$$\begin{aligned} p_i&=(0.15)(20)=3.00\,\text{kg·m/s},\\ p_f&=(0.15)(-15)=-2.25\,\text{kg·m/s}. \end{aligned}$$

Therefore, the impulse on the ball is

$$J=p_f-p_i=-2.25-3.00=-5.25\,\text{N·s}.$$

The negative sign means the impulse points left. If contact lasts $0.010\,\text{s}$, the average net force is

$$F_{\text{avg}}=\frac{J}{\Delta t}=\frac{-5.25}{0.010}=-525\,\text{N}.$$

This is the average force during contact, not necessarily the force at every instant. The force may vary substantially over that interval while producing the same total impulse.

Impulse and conservation of momentum

For a system of objects, add the momentum vectors of all its members to find the total momentum. Internal forces—forces the members exert on one another—can change each object’s momentum. But the internal forces occur in equal-and-opposite pairs, so they cancel in the total momentum balance. The system’s total momentum changes only through the net external impulse:

$$\Delta\mathbf{P}_{\text{system}}=\mathbf{J}_{\text{external}}.$$

Thus total momentum is conserved when the net external impulse is zero, or small enough to neglect for the calculation. During a brief collision, a system might be treated as isolated if external forces such as friction or gravity deliver very little impulse compared with the collision forces. Whether that approximation is reasonable depends on the time interval and the situation.

Conservation of momentum does not mean each object keeps its momentum. It means the vector sum for the system stays constant under the stated condition. Nor does momentum conservation by itself say that kinetic energy is conserved; that depends on the type of collision.

Common mistakes

• Using speed instead of velocity: Direction matters. Reversing direction changes the sign of a one-dimensional velocity and can make the momentum change larger than either momentum alone.
• Ignoring other forces: The theorem uses the net force. If several forces act, their vector sum determines the impulse.
• Confusing force with impulse: Force is measured in newtons; impulse is measured in newton-seconds and depends on how long the force acts.
• Applying conservation to one object: Momentum conservation usually refers to a system. An individual object’s momentum changes when it receives an impulse from another object or the surroundings.
• Assuming every collision conserves kinetic energy: Momentum can be conserved in an isolated collision even when kinetic energy is converted into heat, sound, or deformation.