Momentum
Momentum and Newton's 2nd Law
Extended (Supplement)
Defining momentum
The momentum of a moving object is the product of its mass and its velocity.
\[ p = mv \]
| Symbol | Quantity | Unit |
|---|---|---|
| \( p \) | Momentum | kilogram-metre per second, kg·m/s |
| \( m \) | Mass | kilogram, kg |
| \( v \) | Velocity | metre per second, m/s |
Momentum is a vector quantity — it has both magnitude and direction, the same direction as the velocity. A negative momentum simply means the object is moving in the opposite direction to whatever we have defined as positive.
| Object | Mass (kg) | Velocity (m/s) | Momentum (kg·m/s) |
|---|---|---|---|
| Tennis ball | 0.058 | 60 | 3.5 |
| Car | 1200 | 30 | 36 000 |
| Lorry | 8000 | 25 | 200 000 |
| Stationary object | any | 0 | 0 |
Momentum as a vector
When two objects move in opposite directions, assign a positive direction (usually to the right) and give objects moving left a negative velocity. Then:
- Rightward momentum: positive
- Leftward momentum: negative
Example 1 — calculate momentum
A car of mass 900 kg travels east at 20 m/s. Find its momentum.
\( p = mv = 900 \times 20 = 18\,000\,\text{kg·m/s east} \)
Example 2 — find velocity from momentum
A ball has a momentum of 4.8 kg·m/s and a mass of 0.40 kg. Find its speed.
\( v = \dfrac{p}{m} = \dfrac{4.8}{0.40} = 12\,\text{m/s} \)
Example 3 — change in momentum
A 0.50 kg ball moving at +8.0 m/s bounces off a wall and returns at −8.0 m/s. Find the change in momentum.
Initial momentum: \( p_i = 0.50 \times (+8.0) = +4.0\,\text{kg·m/s} \)
Final momentum: \( p_f = 0.50 \times (-8.0) = -4.0\,\text{kg·m/s} \)
Change: \( \Delta p = p_f - p_i = -4.0 - (+4.0) = -8.0\,\text{kg·m/s} \) (i.e. 8.0 kg·m/s toward the wall)
Link to Newton's Second Law
Newton's Second Law can be written in terms of momentum:
\[ F = \frac{\Delta p}{\Delta t} = \frac{m(v - u)}{t} \]
The resultant force on an object equals the rate of change of its momentum. Since \( \Delta p = m\Delta v \) and \( a = \Delta v / t \), this reduces to \( F = ma \) for constant mass — both forms are equivalent.
The unit kg·m/s and N·s are equivalent (since 1 N = 1 kg·m/s²). Either can be used as the unit of momentum or impulse — the mark scheme accepts both.