Momentum
Momentum (p) is "mass in motion":
p = mv
- Units: kg·m/s
- Momentum is a vector; it has the same direction as velocity.
A slow-moving train can have huge momentum because of its enormous mass. A bullet can have large momentum because of its high speed.
| Object | Mass | Velocity | Momentum |
|---|---|---|---|
| Hockey puck | 0.17 kg | 40 m/s | 6.8 kg·m/s |
| Football player | 100 kg | 8 m/s | 800 kg·m/s |
| Car | 1500 kg | 25 m/s | 37 500 kg·m/s |
Impulse
To change an object’s momentum, a force must act over time. This is impulse:
J = FΔt = Δp = mv_f − mv_i
This comes directly from Newton’s second law: F = ma = mΔv ÷ Δt.
Using impulse for safety
For a given change in momentum, a longer impact time means a smaller force.
| Safety feature | How it works |
|---|---|
| Airbags | Increase stopping time, reducing force on the head |
| Crumple zones in cars | Crush slowly, extending collision time |
| Bending knees when landing | Increases the time to stop |
| Helmets and padding | Compress and spread the impact over more time |
| "Giving" with your hands when catching | Extends catching time |
Example: A 70 kg person stops from 15 m/s.
- Δp = 70 × 15 = 1050 kg·m/s
- Without an airbag (Δt = 0.01 s): F = 1050 ÷ 0.01 = 105 000 N
- With an airbag (Δt = 0.1 s): F = 1050 ÷ 0.1 = 10 500 N, ten times smaller
Follow-through in sports
Golfers, tennis players and batters follow through to increase contact time, increasing the impulse and sending the ball faster.
Conservation of momentum
In an isolated system (no net external force), the total momentum is conserved:
total p before = total p after
m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f
This follows from Newton’s third law: during a collision, objects exert equal and opposite forces on each other for the same time, so their momentum changes are equal and opposite.
Types of collisions
| Type | Momentum | Kinetic energy | Example |
|---|---|---|---|
| Elastic collision | Conserved | Conserved | Billiard balls (nearly), gas molecules |
| Inelastic collision | Conserved | Not conserved (some becomes heat, sound, deformation) | Car crash, football tackle |
| Perfectly inelastic | Conserved | Maximum KE lost; objects stick together | Two train cars coupling |
Example: perfectly inelastic collision
A 1000 kg car moving at 20 m/s hits a stationary 1500 kg car, and they stick together.
- p before = 1000 × 20 + 1500 × 0 = 20 000 kg·m/s
- p after = (2500) v
- v = 20 000 ÷ 2500 = 8 m/s
KE before = ½(1000)(20²) = 200 000 J
KE after = ½(2500)(8²) = 80 000 J
120 000 J was transformed into heat, sound and bent metal.
Example: explosion (recoil)
A 4 kg rifle fires a 0.01 kg bullet at 400 m/s. Total momentum before = 0.
- 0 = (0.01)(400) + (4)(v)
- v = −1 m/s (the rifle recoils backward at 1 m/s)
Rockets work the same way: they push exhaust backward and gain forward momentum, even in empty space.
Two-dimensional collisions
In 2-D, momentum is conserved separately in the x- and y-directions. Billiard players and accident investigators use this to work out how objects were moving before a collision.
Common mistakes
- Forgetting direction. Momentum to the left is negative if right is positive.
- Thinking kinetic energy is always conserved. Only in elastic collisions.
- Thinking momentum is conserved when external forces act, like friction over a long time.
Try it yourself
- A 0.5 kg ball moving at 6 m/s bounces back at 4 m/s. What impulse acted on it?
- A 60 kg skater at rest pushes a 40 kg skater away at 3 m/s. What is the first skater’s velocity?
- A 2 kg ball at 5 m/s hits a 3 kg ball at rest. They stick. Find their final speed and the kinetic energy lost.
(Answers: 5 kg·m/s opposite to the original motion; 2 m/s in the opposite direction; 2 m/s, 25 J − 10 J = 15 J lost)
Quick recap
- Momentum p = mv is a vector.
- Impulse J = FΔt = Δp; longer impact time means smaller force.
- Total momentum is conserved in isolated systems.
- Elastic collisions conserve kinetic energy; inelastic ones don’t.
- Rockets and recoil are momentum conservation in action.