What Is Impulse and How Is It Related to Change in Momentum?

Impulse is the product of the average resultant force acting on a system and the time interval over which it acts. It equals the system's change in momentum: $\vec{J}=\Delta\vec{p}$. This relationship is required in IB Physics A.2 for both SL and HL.

The Relationship

For a constant or average resultant force,

$$\vec{J}=\vec{F}_{\text{avg}}\Delta t=\Delta\vec{p}=\vec{p}_{f}-\vec{p}_{i}.$$

Since momentum is $\vec{p}=m\vec{v}$, for constant mass:

$$\vec{J}=m\vec{v}_{f}-m\vec{v}_{i}=m\Delta\vec{v}.$$

Impulse and momentum are vector quantities, so direction and signs matter. The SI unit of impulse is $\text{N s}$, equivalent to $\text{kg m s}^{-1}$.

Newton's second law gives

$$\vec{F}_{\text{avg}}=\frac{\Delta\vec{p}}{\Delta t}.$$

Multiplying by $\Delta t$ gives $\vec{F}_{\text{avg}}\Delta t=\Delta\vec{p}$. If force varies with time, impulse is the . A longer collision time therefore produces the same momentum change with a smaller average resultant force, which explains the operation of airbags and crumple zones.

Worked Example

A $0.20\,\text{kg}$ ball moves right at $6.0\,\text{m s}^{-1}$ and rebounds left at $4.0\,\text{m s}^{-1}$. Taking right as positive:

$$J=\Delta p=m(v_f-v_i)=0.20(-4.0-6.0)=-2.0\,\text{N s}.$$

The negative sign means the impulse is directed left. Notice that the momentum change is not $0.20(6.0-4.0)$: reversing direction requires signed velocities. If contact lasts $0.050\,\text{s}$,

$$F_{\text{avg}}=\frac{J}{\Delta t}=\frac{-2.0}{0.050}=-40\,\text{N}.$$

A common misconception is that impulse equals force. It does not: impulse depends on both resultant force and contact time. Another misconception is that momentum is conserved for the ball alone during impact; its momentum changes because an external force acts on it.

During a collision, the forces on the two interacting bodies form a Newton's third-law pair. They act for the same time and produce equal-magnitude, opposite impulses, so the total momentum of an isolated two-body system remains constant even though each body's momentum changes.

Exam Technique

For Calculate questions, state a positive direction, write $J=\Delta p$, substitute signed velocities, show each stage, and give the direction of the final vector. For force-time graphs, calculate the signed area, not the gradient. Use the data booklet equation and distinguish the resultant force from any single force acting on the system.