14 4.6 Life applications of the second law (force, time and change of momentum)
From F = Δ(mv)/Δt, i.e. F·Δt = Δ(mv) (the product Ft is called the impulse), the force needed to bring a given moving body to rest depends on the time it takes:
- If the change of momentum happens over a longer time, the force is smaller (the effect of the impact is less).
- If it happens in a shorter time, the force is larger.
This explains:
- A car hitting a wall is more destructive than a car hitting a heap of straw: the wall stops it in a very short time (large force); straw takes longer (smaller force).
- A person falling from a height into water is not hurt but falling onto ground may hurt: water brings him to rest more slowly.
- The severity of the injury increases with the height of the fall, because the velocity (and so the momentum) at impact increases.
- An egg does not break when it falls on a cushion, but it breaks on the floor.
- A collision of a large truck with a wall is more destructive than that of a small truck: larger mass → larger momentum → larger force for the same stopping time.
- Airbags in cars increase the time over which the driver's body comes to rest in a collision, reducing the force on him. Similarly, a cricket or football goalkeeper pulls the hands back while catching a fast ball to lengthen the stopping time.
✍️ Worked examples
Example 6 (numerical). A 0.15 kg ball moving at 20 m/s is caught and stopped.
- Stopped in 0.01 s (hard hands): F = 0.15 × 20/0.01 = 300 N.
- Stopped in 0.10 s (hands drawn back): F = 0.15 × 20/0.10 = 30 N, ten times smaller.
Example 7 (airbag). A driver of mass 70 kg moving at 14 m/s is stopped in 0.05 s by a rigid steering wheel: F = 70 × 14/0.05 = 19 600 N. With an airbag extending the time to 0.25 s: F = 70 × 14/0.25 = 3 920 N.