Conservation of momentum

7 minAS Level4.3.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can apply conservation of linear momentum to two bodies colliding in a straight line, including the case where they stick together, and use signs correctly when they approach from opposite directions.

Key to the symbols

Symbol	Meaning
$m_1$ , $m_2$	masses of the two objects, in kg
$u_1$ , $u_2$	their velocities before the collision, in $\text{m s}^{-1}$
$v_1$ , $v_2$	their velocities after the collision
$v$	the shared velocity if they stick together

Coalesce means "stick together and move as one".

Two trolleys and a click

Line up two trolleys on a track. Give one a push so it hits the other, and watch what happens afterwards.

During the collision each trolley pushes the other. Those forces are equal and opposite (Newton's third law) and last for the same time, so the momentum one loses is exactly the momentum the other gains. The total does not change.

That is conservation of momentum, and it turns a messy collision into a single equation.

Definition

Conservation of linear momentum

A system is the group of bodies you are studying, here the two colliding bodies. If no external resultant force (a force from outside the system) acts on it, the total momentum of the system stays constant. For two bodies colliding along a line: $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2,$ where $u$ denotes velocity before the impact and $v$ after.

The total momentum is 6 kg m s⁻¹ before and after; the mass has changed, so the velocity must change too.

The coalescing case

If the two bodies stick together, they move off as one object of mass $m_1 + m_2$ with a single velocity $v$ . The equation becomes

$m_1 u_1 + m_2 u_2 = (m_1 + m_2)v,$

$v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}.$

This is the "particles coalesce on impact" case in the syllabus.

Notice what the formula does: it is a weighted average of the two velocities. Each velocity is weighted by its mass, so the result is pulled towards the velocity of the heavier body.

Worked example: Two trolleys that stick together

A trolley of mass $3.0\ \text{kg}$ moves at $4.0\ \text{m s}^{-1}$ towards a stationary trolley of mass $1.0\ \text{kg}$ . They stick together on impact.

Find their common velocity afterwards.

Have a go first?

Worked example: Approach from opposite directions

A $3.0\ \text{kg}$ trolley moves at $4.0\ \text{m s}^{-1}$ to the right. A $1.0\ \text{kg}$ trolley moves at $2.0\ \text{m s}^{-1}$ to the left. They stick together.

Find their common velocity after the impact.

Have a go first?

A routine that never fails

Draw a before-and-after sketch, with an arrow for every velocity.
Choose a positive direction and write it down.
Convert every velocity to a signed number.
Write "momentum before = momentum after".
If the bodies coalesce, use one combined mass on the right.
Solve, then interpret the sign of the answer as a direction.

Check your answer for plausibility: the final velocity of a coalescing pair must lie between the two initial velocities, because it is a weighted average.

Common mistakes

Momentum is conserved only if the bodies stick together.

It is conserved in every impact where no external resultant force acts, whether they stick, bounce or separate.

You can add the speeds even when the bodies move in opposite directions.

Velocities are signed. A body moving the other way has negative momentum, which can reduce or even cancel the total.

Kinetic energy is always conserved in a collision.

Momentum is conserved, but kinetic energy (the energy of motion, covered in topic 4.5) is usually lost to sound, heat and squashing. When bodies coalesce, some kinetic energy is always lost.

The heavier body always continues in its original direction.

In the exam

Write the conservation equation explicitly with the velocities substituted, signs included. The method mark is for the equation, not just the final number.

If the question says "they coalesce" or "stick together", use a single combined mass on the right-hand side.

Give the direction of the final velocity, either in words or by the sign with your convention stated.

Check yourself

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Stretch

Answer the 2 questions above to continue.