Force from impulse

Data

• Alice´s mass: m = 55kg
• Velocity before the impact: v_i = 90 km/h = 90·1000/3600 = 25 m/s
• Duration of the impact: t = 0.05 s

Previous considerations

Observe when we say Alice "weights" 55 kg, is just a popular way of saying that the mass of Alice is 55 kg, since the weight is, actually, a force.

Resolution

Impulse-momentum theorem connects the variation of the linear momentum to the impulse itself:

$$\overrightarrow{J}=\overrightarrow{F}·∆t=∆\overrightarrow{p}\Rightarrow J=F·∆t=∆p$$

We know that, after the accident, the final velocity is zero, so we can write:

$$∆p=p_f-p_i=\cancel{m·v_f}-m·v_i=-55·25=-1375 kg·m/s$$

That variation coincides numerically with the impulse, thus we can write:

$$J=F·∆t=∆p\Rightarrow F=\frac{∆p}{∆t}=\frac{-1375}{0.05}=-27500 N$$

Where the sign - indicates the force is against the initial sense of the motion. Observe the relation   $$F=\frac{∆p}{∆t}$$ belongs to the Newton´s second law, from which we could have started to resolve the problem without the need to recur to the impulse.