# 3.2.3 Impulse-momentum Theorem (J=FΔt=Δp)

Starting from $F=\frac{{\Delta p}}{{\Delta t}}$, we can re-arrange the terms to get

$F\Delta t=\Delta p$

where F could represent either a constant force or the average force for the duration Δt.

The quantity $F\Delta t$ is given the name impulse. It is usually denoted by the symbol J and has the unit N s.

The following equation is called the impulse-momentum theorem.

$J=F\Delta t=\Delta p$

Quite simply, it says that the application of a force F (constant or average) for a duration of time $\Delta t$ causes a momentum change of $\Delta p$.

For example, to increase a mass’s momentum from 0 kg m s-1 to 12 kg m s-1 requires an impulse of 12 N s. This impulse can be delivered by exerting a force of 1 N for 12 s on the mass, or 2 N for 6 s, or 3 N for 4 s, or 8 N for 1.5 s,… because all of them correspond to an impulse of 12 N s.

Buying time to reduce impact force

The concept of impulse is very helpful in understanding why gymnasts always bend their knees when they land. To come to a rest, a gymnast must lose all the downward momentum. This requires an upward impulse. Whether the bent knees are bent or straight, the required momentum change (and thus the required impulse) during the landing is the same. By bending the knees, however, gymnasts extend the duration of impact Δt. This allows them to achieve the landing with a smaller impact force F. In short, the required $\Delta p=J$ is the same. But small F big Δt is a graceful touch-down whereas big F small Δt is a bone-shattering experience.

The same physics applies for anti-crush devices such as air bags, car crumple zones, safety nets, stuntman boxes, corrugated paper, bubble wraps. The required impulse J is fixed, so it is always about extending the impact duration Δt to reduce the impact force F.

Demonstrations

Bird Egg

Gymnasts and Air Bags

Concept Test