Momentum

(Higher Tier only) The whole of this topic, momentum, is assessed at Higher Tier only in AQA GCSE Physics. Foundation students do not need it.

Every moving object has momentum: a measure of how hard it is to stop. A heavy lorry moving slowly and a light bullet moving fast can both carry large momentum. Momentum, given the symbol $p$, is defined as mass multiplied by velocity:

$p=mv$

• $p$ is momentum in kilograms metres per second (kg m/s)
• $m$ is mass in kilograms (kg)
• $v$ is velocity in metres per second (m/s)

Because velocity is a vector (it has direction), momentum is also a vector. It has both a size and a direction. When objects move in opposite directions, one direction is taken as positive and the other as negative.

$p=mv$ is a recall-and-apply equation. You must recall and apply it; it is not given on the equation sheet.

Worked example — a 1500 kg car travels at 12 m/s. Find its momentum.

$p=mv=1500\times 12=18,000\text{ kg m/s}$

In a closed system (one with no external resultant force acting), the total momentum stays constant. This is the conservation of momentum:

$\text{total momentum before}=\text{total momentum after}$

This applies to any event where objects interact, such as collisions and explosions, as long as the system is closed. To use it, add up the momentum of every object before the event and set it equal to the total momentum afterwards.

In a closed system, the total momentum before an event is equal to the total momentum after the event. Momentum is always conserved when no external resultant force acts.

Direction matters because momentum is a vector. Choose one direction as positive before you start, and give objects moving the other way a negative velocity. Keeping the signs consistent is the key to getting collision questions right.

(Separate Physics only) Collision and explosion calculations are assessed in separate Physics, not in Combined Science. They are also Higher Tier only.

When two objects collide and move off together, treat the joined objects as one combined mass afterwards.

Worked example — a 2.0 kg trolley moving at 3.0 m/s collides with a stationary 1.0 kg trolley. They stick together. Find their common velocity after the collision.

Step 1 — total momentum before:

$p_{\text{before}}=(2.0\times 3.0)+(1.0\times 0)=6.0\text{ kg m/s}$

Step 2 — after the collision the combined mass is $2.0+1.0=3.0$ kg, moving at velocity $v$:

$p_{\text{after}}=3.0\times v$

Step 3 — apply conservation of momentum ($p_{\text{before}}=p_{\text{after}}$):

$6.0=3.0\times v\Rightarrow v=\frac{6.0}{3.0}=2.0\text{ m/s}$

The combined trolleys move off at 2.0 m/s in the original direction.

An explosion is the reverse of a sticking collision: one object at rest separates into parts that move apart. Before the explosion the total momentum is zero, so the total momentum afterwards must also be zero. The parts must move in opposite directions so their momenta cancel.

Worked example — a stationary cannon of mass 400 kg fires a 4.0 kg cannonball forwards at 60 m/s. Find the recoil velocity of the cannon.

Step 1 — total momentum before is zero (everything at rest):

$p_{\text{before}}=0$

Step 2 — total momentum after must also be zero. Take forwards as positive:

$(4.0\times 60)+(400\times v)=0$

Step 3 — solve for the cannon's velocity $v$:

$240+400v=0\Rightarrow v=\frac{-240}{400}=-0.6\text{ m/s}$

The negative sign shows the cannon recoils backwards at 0.6 m/s, opposite to the ball.

When a resultant force acts on a moving object, its momentum changes. The force equals the rate of change of momentum:

$F=\frac{m\Delta v}{\Delta t}$

• $F$ is the resultant force in newtons (N)
• $m\Delta v$ is the change in momentum in kg m/s
• $\Delta t$ is the time taken for the change in seconds (s)

(Separate Physics only) This equation and its use for safety features are separate Physics content, Higher Tier only.

$F=\frac{m\Delta v}{\Delta t}$ is given on the Physics equation sheet. You do not have to recall it, but you must be able to use it.

The equation shows a key idea: for a given change in momentum, a longer time gives a smaller force. This is the physics behind almost every vehicle safety feature.

Worked example — a 60 kg passenger moving at 15 m/s is brought to rest. Compare the force with and without a seat belt.

Without a belt, the passenger stops in 0.05 s against the dashboard:

$F=\frac{60\times 15}{0.05}=\frac{900}{0.05}=18,000\text{ N}$

With a belt that stretches, the passenger stops in 0.5 s:

$F=\frac{60\times 15}{0.5}=\frac{900}{0.5}=1800\text{ N}$

The same change in momentum spread over ten times the time gives one tenth of the force.

Vehicle and sports safety features all work the same way: they increase the time taken to change a person's momentum, which reduces the force on the body, following $F=m\Delta v/\Delta t$.

Safety feature	How it reduces force
Air bag	Inflates so the head decelerates over a longer time and distance
Seat belt	Stretches slightly, increasing the stopping time
Crumple zone	Front of the car folds, extending the collision time
Crash mat	Compresses, increasing the time to stop a falling person
Cushioned surface (playground)	Squashes, lengthening the stopping time
Cycle helmet	Foam crushes, spreading the impact over more time

Every one of these features increases the collision time, which lowers the rate of change of momentum and so lowers the force. State the time-then-force chain explicitly for full marks.

1. Forgetting direction signs

Momentum is a vector. Give objects moving in opposite directions opposite signs before adding. A common error is treating a recoil or head-on collision as if all momenta were positive.

2. Using speed instead of velocity in the wrong direction

In a head-on collision, the two objects have velocities of opposite sign. Plugging both in as positive gives the wrong total momentum before the collision.

3. Saying safety features reduce the change in momentum

Air bags and seat belts do not change the momentum a person must lose; that is fixed by their mass and speed. They increase the time, which reduces the force.

4. Wrong units for momentum

Momentum is in kg m/s, not newtons and not kg. Newtons are the unit of force.

5. Forgetting the combined mass after a sticking collision

When objects join, the mass after the collision is the sum of the masses. Using only one object's mass gives the wrong final velocity.