Impulse and change in momentum

Newton's second law written with momentum

• A force acting for a time changes an object's momentum. Newton's second law can be written:

$$F = \frac{\Delta p}{t}$$

• $F$ — the (average) net force acting (N)

• $\Delta p$ — change in momentum (kg m s⁻¹)

• $t$ — time for which the force acts (s)

• Rearranged, this gives the impulse:

$$Ft = \Delta p = mv_f - mv_i$$

• Impulse is the product of force and the time it acts for.
• Impulse has units of N s, which are identical to kg m s⁻¹ — impulse and change in momentum are the same physical quantity.
• Impulse is a vector, in the direction of the force.

Why this is the same as $F = ma$

• Starting from $F = \Delta p / t$ with a constant mass:

$$F = \frac{mv_f - mv_i}{t} = m\frac{v_f - v_i}{t} = ma$$

• So $F = ma$ and $F = \Delta p / t$ are the same law. Use whichever fits the information you have:
  • given masses and velocities → the momentum form,
  • given a mass and an acceleration → $F = ma$.

The force–time trade-off

For a fixed change in momentum, force and time are inversely proportional:

$$F = \frac{\Delta p}{t}$$

• Longer contact time → smaller force.
• Shorter contact time → larger force.
• The change in momentum is decided by the situation — how fast the object was going and how fast it ends up. Only the time can be engineered, and that is what every safety device does.

Safety devices explained with impulse

Every one of these works the same way: the momentum change is fixed, so extending the time reduces the force.

• Crumple zones — the car body deforms progressively, so the passenger comes to rest over a longer time rather than instantaneously.
• Airbags — the head decelerates over the depth of the bag instead of over the few millimetres of a steering wheel.
• Seatbelts — designed to stretch slightly, spreading the deceleration over more time (and over a larger area of the body).
• Helmets — the crushable foam liner extends the stopping time of the skull.
• Crash mats and gym landing pits — compress under a falling gymnast.
• Bending your knees when you land — increases the stopping distance and so the stopping time.

A complete exam answer names all three steps:

• State that the change in momentum is the same either way, because the mass and the velocities are unchanged.
• State that the device increases the contact time.
• Conclude from $F = \Delta p / t$ that the force on the person is therefore smaller, reducing injury.