Momentum

Force as the rate of change of momentum

• Combining F = ma and the definition p = mv gives a more general form of Newton's second law:

F = Δp / t = (mv − mu) / t

• where:
  • F = resultant force (N)
  • mv − mu = change in momentum, Δp (kg m/s), i.e. final momentum minus initial momentum
  • t = time over which the change happens (s)
• In words: the resultant force on an object equals its rate of change of momentum

Contact time and impact force

• For a given change in momentum, the force and the contact time are inversely proportional to one another:
  • Halve the contact time and the impact force doubles
  • Triple the contact time and the impact force falls to a third
• Real safety design therefore aims to extend the contact time during a collision to make the impact force as small as possible (see section 5)

Example A — during the same return shot, a 0.40 kg football undergoes the same change in momentum of 8.0 kg m/s in two different scenarios: scenario 1 has a foot-on-ball contact time of 0.050 s, scenario 2 has a contact time of 0.50 s. Compare the average force on the ball.

• Scenario 1: F = Δp / t = 8.0 / 0.050 = 160 N
• Scenario 2: F = Δp / t = 8.0 / 0.50 = 16 N
• The same change in momentum, delivered over ten times longer, gives a force that is ten times smaller

Example B — a 1200 kg car drives at 18 m/s into a wall and rebounds at 4.0 m/s in the opposite direction. The collision lasts 0.20 s. Calculate the average force on the car and state its direction.

• Take the car's forward (towards-wall) direction as positive: u = +18 m/s, v = −4.0 m/s
• Δp = m(v − u) = 1200 × (−4.0 − 18) = 1200 × (−22) = −26 400 kg m/s
• F = Δp / t = −26 400 / 0.20 = −132 000 N
• The minus sign tells you the force on the car points in the opposite direction to its original motion, that is, backwards from the wall, as you would expect