Linear momentum

6 minAS Level4.3.1

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

By the end of this lesson

By the end you can calculate the momentum of a particle, add momenta as vectors in one dimension, and find the change in momentum when a velocity reverses.

Key to the symbols

Symbol	Meaning	Unit
$p$	momentum	$\text{kg m s}^{-1}$
$m$	mass	kg
$u$	velocity before (at the start)	$\text{m s}^{-1}$
$v$	velocity after (at the end)	$\text{m s}^{-1}$
$m_1, v_1$ and $m_2, v_2$	mass and velocity of the first and second objects
$\Delta$	"the change in", so $\Delta p$ means the change in momentum

Why a lorry hurts more than a bicycle

A bicycle and a lorry can move at the same speed, but the lorry is far harder to stop.

What matters is not speed alone, nor mass alone, but the two multiplied together.

That product is momentum: a measure of "quantity of motion". Doubling the mass doubles it; doubling the speed doubles it; doing both multiplies it by four.

Definition

Linear momentum

The momentum of a particle is its mass multiplied by its velocity: $p = mv.$ It is a vector in the direction of the velocity, measured in $\text{kg m s}^{-1}$ .

One dimension, one sign

In this course all motion is in a straight line, so momentum is handled with signs.

Choose a positive direction, then:

a particle moving that way has positive momentum;
a particle moving the other way has negative momentum.

The total momentum of several particles is the vector sum:

$p_{\text{total}} = m_1 v_1 + m_2 v_2 + \cdots$

Two equal masses moving towards each other at the same speed have momenta $+mv$ and $-mv$ , so the total is zero. Nothing is lost or cancelled in a mysterious way: the momenta simply point in opposite directions.

Change in momentum

When the velocity changes, the momentum changes. The change is

$\Delta p = mv - mu = m(v - u),$

where $u$ is the velocity before and $v$ after. The order matters, and so do the signs.

A ball hitting a wall and bouncing back has $u$ and $v$ in opposite directions. If the direction towards the wall is taken as positive, then $u$ is positive and $v$ is negative. So the size of the change is the sum of the two momenta, not their difference (in the example below, $1.2 + 0.80 = 2.0$ ).

Worked example: A ball bouncing off a wall

A ball of mass $0.20\ \text{kg}$ hits a wall at $6.0\ \text{m s}^{-1}$ and rebounds at $4.0\ \text{m s}^{-1}$ . Take the direction towards the wall as positive.

Find the change in momentum of the ball.

Have a go first?

Common mistakes

Momentum is a scalar because it is just mass times speed.

It uses velocity, not speed, so it carries a direction. Opposite momenta can cancel in a total.

A change in momentum is always positive.

$\Delta p = m(v-u)$ can be negative; the sign gives the direction of the change.

In the exam

State your positive direction and give momenta with signs. Examiners accept a negative answer as a direction, but they cannot follow arithmetic in which the sign convention changes halfway.

For a reversal, write $u$ and $v$ with opposite signs explicitly. This is the single most common error in the topic.

Quote the unit $\text{kg m s}^{-1}$ and do not confuse momentum with force or energy.

Check yourself

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.