🐘 Big mass, low velocity
An elephant of mass 4000 kg walks at 2 m/s.
p = 4000 × 2 = 8000 kg m/s
It is slow, but its huge mass gives it a lot of momentum.
Forces, Movement, Shape and Momentum » Momentum
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 1.25P, 1.2P
- What momentum is and the equation momentum = mass × velocity
- The unit of momentum, the kilogram metre/second (kg m/s)
- Why momentum has a direction and can be negative
- How to rearrange the equation to find mass or velocity
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Unlock This CourseWhat is momentum?
This lesson is examined in Paper 2 only.
Which is harder to stop: a bowling ball rolling towards you or a table tennis ball moving at the same velocity? The bowling ball, because it has far more mass. Now think of two identical bowling balls. The faster one is harder to stop.
Physicists put these two ideas together in one quantity called momentum. Every moving object has momentum. The more mass it has and the faster it moves, the more momentum it has.
momentum = mass × velocity
p = m × v
You must learn this equation. It is not given on the formulae page of the exam paper. Notice that the symbol for momentum is a small p, not an m. The letter m is already used for mass.
| Quantity | Symbol | Unit |
|---|---|---|
| Momentum | p | kilogram metre/second (kg m/s) |
| Mass | m | kilogram (kg) |
| Velocity | v | metre/second (m/s) |
Key terms:
- Momentum (p): the mass of an object multiplied by its velocity.
- Kilogram metre/second (kg m/s): the unit of momentum.
The unit of momentum
The unit of momentum comes straight from the equation. Mass is in kilograms and velocity is in metres per second. Multiply them and you get kilogram metres per second, written kg m/s.
The unit does not have a special name of its own, so you just have to remember it. If you forget it in the exam, write down p = m × v and multiply the units: kg × m/s = kg m/s.
Mass and velocity both matter
🏒 Small mass, high velocity
An ice hockey puck of mass 0.17 kg flies at 30 m/s.
p = 0.17 × 30 = 5.1 kg m/s
It is fast, but its small mass means it has much less momentum than the elephant.
Momentum is directly proportional to velocity. If the same object doubles its velocity, its momentum doubles. Momentum is also directly proportional to mass. At the same velocity, an object with twice the mass has twice the momentum.
An object that is not moving has no momentum. A parked lorry has a mass of thousands of kilograms, but its velocity is 0 m/s, so its momentum is 0 kg m/s.
Using the equation
You can rearrange p = m × v to find any one of the three quantities:
- to find the momentum: p = m × v
- to find the mass: m = p / v
- to find the velocity: v = p / m
In every calculation write the equation, put in the numbers, then give the answer with its unit.
Worked example 1: finding the momentum
A bowling ball of mass 6 kg rolls down a lane at 5 m/s. Calculate its momentum.
p = m × v
p = 6 × 5
p = 30 kg m/s
Worked example 2: finding the mass
A dog runs across a park at 6 m/s. Its momentum is 150 kg m/s. Calculate the mass of the dog.
m = p / v
m = 150 / 6
m = 25 kg
Worked example 3: finding the velocity
A canoe and its paddler have a total mass of 100 kg. Their momentum is 250 kg m/s. Calculate their velocity.
v = p / m
v = 250 / 100
v = 2.5 m/s
Check your units first. The mass must be in kilograms. A mass in grams must be divided by 1000.
Worked example 4: mass in grams
An arrow of mass 50 g leaves a bow at 60 m/s. Calculate its momentum.
Mass = 50 / 1000 = 0.05 kg
p = m × v
p = 0.05 × 60
p = 3 kg m/s
Momentum has a direction
Remember, velocity is a vector: it has a size and a direction. Mass is a scalar. When you multiply a scalar by a vector you get a vector, so momentum is a vector quantity. The momentum of an object is always in the same direction as its velocity.
When objects move along a straight line, we show direction with plus and minus signs. Follow these steps:
- Choose one direction to be positive, for example to the right. Say which one you have chosen.
- Anything moving that way has a positive velocity and a positive momentum.
- Anything moving the opposite way has a negative velocity and a negative momentum.
➕ Moving to the right
A rugby player of mass 90 kg runs to the right at 6 m/s. Take right as positive.
p = 90 × 6
p = +540 kg m/s
➖ Moving to the left
Another player of mass 100 kg runs to the left at 5 m/s. Her velocity is −5 m/s.
p = 100 × (−5)
p = −500 kg m/s
A negative momentum is not a "small" momentum. The minus sign only tells you the direction. The second player has 500 kg m/s of momentum to the left.
Worked example 5: a change of direction
A handball of mass 0.4 kg hits a wall at 10 m/s and bounces straight back at 8 m/s. Calculate its momentum before and after it hits the wall.
Take the direction towards the wall as positive.
Before: p = m × v = 0.4 × 10 = +4 kg m/s
After: the ball is moving away from the wall, so v = −8 m/s.
p = 0.4 × (−8) = −3.2 kg m/s
The sign has changed because the direction has changed.
Getting the signs right matters a great deal when objects collide. You will use them in the lesson Conservation of Momentum.
Common mistakes
- Using the wrong symbol. Momentum is p. Writing m = m × v makes no sense.
- Wrong or missing unit. Momentum is in kg m/s. It is not in N, kg or m/s.
- Leaving mass in grams. Change grams to kilograms before you multiply.
- Forgetting the direction. If two objects move in opposite directions, one of them must have a negative velocity and a negative momentum.
- Dividing the wrong way round. Mass and velocity are both found by dividing the momentum: m = p / v and v = p / m. Momentum always goes on top.
- Mixing up momentum and force. Momentum is something a moving object has. It is not a push or a pull.
Exam-style question
Two dodgem cars move along the same straight line at a funfair. Take the direction to the right as positive.
(a) State the equation linking momentum, mass and velocity. [1 mark]
(b) Dodgem A and its rider have a total mass of 200 kg. They move to the right at 3 m/s. Calculate the momentum of dodgem A and its rider. Give the unit. [3 marks]
(c) Dodgem B and its rider have a total mass of 250 kg. Their momentum is −500 kg m/s. Calculate the velocity of dodgem B. [3 marks]
(d) State what the minus sign in your answer to part (c) shows. [1 mark]
(e) Dodgem A speeds up to 6 m/s in the same direction. State what happens to its momentum. [1 mark]
Model answer
(a) momentum = mass × velocity, or p = m × v (1)
(b) p = 200 × 3 (1)
p = 600 (1)
kg m/s (1)
(c) v = p / m (1)
v = −500 / 250 (1)
v = −2 m/s (1)
(d) Dodgem B is moving to the left, in the opposite direction to dodgem A. (1)
(e) It doubles, to 1200 kg m/s. (1)
Exam tip
In part (b) the unit has a mark of its own, so learn kg m/s. In part (c) keep the minus sign all the way through your working. If you drop it you get 2 m/s, which says dodgem B is moving to the right, and you lose the final mark.