Impulse

What impulse is

When a force acts on a body for a time $\Delta t$, the product of force and time is called impulse:

$\vec{J} = \vec{F} \cdot \Delta t$

Impulse equals the change in momentum of the body:

$\vec{J} = \Delta\vec{p} = \vec{p}_f - \vec{p}_i$

Derivation from Newton's Second Law

From Newton's Second Law:

$\vec{F} = \frac{d\vec{p}}{dt}$

Rearrange:

$d\vec{p} = \vec{F} \, dt$

Integrate over the time interval from $t_1$ to $t_2$:

$\int_{\vec{p}_i}^{\vec{p}_f} d\vec{p} = \int_{t_1}^{t_2} \vec{F} \, dt$

$\vec{p}_f - \vec{p}_i = \int_{t_1}^{t_2} \vec{F} \, dt$

For a constant force:

$\Delta\vec{p} = \vec{F} \cdot \Delta t = \vec{J}$

Why impulse is useful

Many forces in nature act for a very short time — a bat hitting a ball, a hammer hitting a nail, a collision between cars. During these brief contacts, the force varies rapidly and is difficult to measure at every instant.

But the total effect — the change in momentum — is measurable. Impulse lets us relate the overall change in momentum to the average force without needing to know the force at every instant.

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

The impulse-momentum theorem

$\vec{J} = \Delta\vec{p}$

This is the impulse-momentum theorem. It says: the impulse delivered to a body equals the change in momentum of that body. It follows directly from Newton's Second Law integrated over time.

Variable force — area under F-t graph

When force varies with time, the impulse is the area under the force-time graph:

$\vec{J} = \int_{t_1}^{t_2} \vec{F}(t) \, dt$

For a constant force, this is simply a rectangle: $J = F \cdot \Delta t$.

For a varying force (like during a collision), the area under the curve gives the total impulse.

The trade-off between force and time

Since $J = F \cdot \Delta t = \Delta p$ is fixed (the momentum change is determined by the situation), a longer contact time means a smaller average force.

This is the physics behind:

• Airbags: increase the collision time → reduce the peak force on the passenger
• Catching a cricket ball: pull your hands back as you catch → increase time of contact → reduce force on hands
• Gymnastics mats: increase time of impact with the floor → reduce force on the gymnast
• Car crumple zones: designed to increase collision time → reduce peak deceleration force

Conversely, a shorter time for the same momentum change means a larger force — this is why a karate chop delivers more force than a slow push of the same momentum change.

Units

$[J] = \text{N} \cdot \text{s} = \text{kg} \cdot \text{m/s}$

Same as momentum — consistent with $J = \Delta p$.