Linear momentum

CIE A-Level Physics· Unit 3: Dynamics· 15 min read

1. Linear Momentum and Impulse★★☆☆☆⏱ 3 min

📘 Definition

Linear momentum

\vec{p}

Momentum is the product of an object's mass and velocity. It is a vector quantity, with direction matching the velocity of the object.

Example:

A 3 kg block moving at 2 m s⁻¹ right has momentum $p = 3 \times 2 = 6$ kg m s⁻¹ right.

Impulse describes the effect of a force acting over a period of time to change an object's momentum.

📘 Definition

Impulse

$J$

Impulse equals the product of the average force acting on an object and the contact time, and is equal to the change in momentum of the object: $J = F_{avg} Δ t = Δ p$ .

📐 Worked Example

A 0.15 kg ball travels at 10 m s⁻¹ towards a wall, and rebounds at 8 m s⁻¹ along the same line. Calculate the impulse exerted on the ball by the wall.

1
Define the positive direction as away from the wall. Initial velocity $u = - 10$ m s⁻¹, final velocity $v = + 8$ m s⁻¹.
2
Impulse equals change in momentum $Δ p = m (v - u)$ :
3
$J = 0.15 \left(8 - (-10)\right) = 0.15 \times 18 = 2.7 N s$
4
The impulse exerted on the ball is 2.7 N s away from the wall.

Exam tip:

Always define your positive direction first to avoid sign errors in rebound impulse calculations.

2. Momentum and Newton's Second Law★★☆☆☆⏱ 4 min

Newton's original formulation of the second law of motion is written in terms of momentum, rather than acceleration. This form works for both constant and changing mass.

📘 Definition

Newton's Second Law (momentum form)

The net force acting on an object equals the rate of change of its linear momentum.

Example:

If mass is constant: $F_{net} = \frac{Δ p}{Δ t} = \frac{m Δ v}{Δ t} = ma$ , which matches the standard $F = ma$ form.

📐 Worked Example

A constant net force of 12 N acts on a 4 kg object initially at rest. What is the momentum of the object after 3 seconds?

1
Rearrange Newton's second law to solve for change in momentum:
2
$\Delta p = F_{\text{net}} \Delta t = 12 \times 3 = 36 kg m s^{-1}$
3
Initial momentum is 0, so final momentum equals the change in momentum.
4
Final momentum = $36$ kg m s⁻¹

3. Conservation of Linear Momentum★★★☆☆⏱ 5 min

The principle of conservation of momentum is one of the most widely tested concepts in CIE A-Level dynamics.

📘 Definition

Principle of Conservation of Linear Momentum

For a closed system with no net external force acting, the total linear momentum of the system is constant. This means total momentum before an interaction equals total momentum after.

📐 Worked Example

A 5000 kg truck moving at 12 m s⁻¹ collides with a stationary 1000 kg car. After collision, they move together along the same line. Find their common speed after collision.

1
Calculate total momentum before collision:
2
$p_{before} = m_{truck}u_{truck} + m_{car}u_{car} = (5000 \times 12) + (1000 \times 0) = 60000 kg m s^{-1}$
3
By conservation of momentum, $p_{a f t er} = p_{b e f or e} = 60000$ kg m s⁻¹. Combined mass $= 5000 + 1000 = 6000$ kg. Let $v$ = common speed:
4
$6000 v = 60000 \implies v = 10 m s^{-1}$
5
Common speed after collision is 10 m s⁻¹.

Exam tip:

Always check you add masses correctly for collisions where objects stick together after impact.

4. Elastic and Inelastic Collisions★★★☆☆⏱ 5 min

Collisions are classified based on whether kinetic energy is conserved during the interaction:

Elastic collision: Total momentum and total kinetic energy are both conserved.
Inelastic collision: Only total momentum is conserved; kinetic energy is converted to heat, sound or other forms, so total kinetic energy decreases.
Perfectly inelastic collision: Objects stick together after collision, maximum kinetic energy is lost.

📐 Worked Example

Show that the truck-car collision in the previous example is inelastic by calculating the change in kinetic energy.

1
Calculate total initial kinetic energy before collision:
2
$KE_{initial} = \frac{1}{2}m_{truck}u_{truck}^2 = 0.5 \times 5000 \times 12^2 = 360000 J$
3
Calculate total final kinetic energy after collision:
4
$KE_{final} = \frac{1}{2}(m_{truck} + m_{car})v^2 = 0.5 \times 6000 \times 10^2 = 300000 J$
5
60000 J of kinetic energy is lost during the collision, so kinetic energy is not conserved. Therefore the collision is inelastic.

5. Common Pitfalls

Wrong move:

Forgetting momentum is a vector, so ignoring sign when adding momenta for opposite directions

Why:

Failing to account for direction gives an incorrect total momentum for the system

Correct move:

Always define a positive direction before starting calculations, assign negative signs to momentum moving in the opposite direction

Wrong move:

Assuming kinetic energy is conserved in all collisions, just like momentum

Why:

Many students mix up the conservation rules for momentum and kinetic energy

Correct move:

Momentum is always conserved (no external forces). Kinetic energy is only conserved for elastic collisions

Wrong move:

Calculating rebound impulse as $m (v + u)$ with both speeds positive without direction convention

Why:

This can lead to magnitude errors if the direction is not accounted for correctly

Correct move:

Always use $Δ p = m (v - u)$ with correct signed values for initial and final velocity

Wrong move:

Applying conservation of momentum when external forces like friction are acting on the system

Why:

The principle only holds for closed systems with no net external force

Correct move:

Only use conservation of momentum if no external forces act, or the interaction time is too short for external forces to change total momentum

6. Quick Reference Cheatsheet

Quantity/Concept	Formula	Key Fact
Linear momentum	$p = m v$	Vector, units kg m s⁻¹
Impulse	$J = F Δ t = Δ p$	Equal to change in momentum
Newton's 2nd Law	$F_{n e t} = \frac{Δ p}{Δ t}$	Original, general form of the law
Conservation of momentum	$\sum p_{b e f or e} = \sum p_{a f t er}$	Holds for closed systems, no external forces
Elastic collision		Momentum and kinetic energy conserved
Inelastic collision		Only momentum conserved; KE lost

7. Frequently Asked

Can momentum be negative?

Yes, momentum is a vector. Negative values indicate momentum acting opposite to your defined positive direction. Always state your direction convention in calculations.

Is momentum always conserved in collisions?

Yes, as long as no net external force acts on the system. Only kinetic energy is not conserved for inelastic collisions.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Impulse for a rebounding ball
2023 · 2
Momentum conservation in collision
2024 · 1
Classify elastic/inelastic collision

Going deeper

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What's Next

Linear momentum is a core foundational concept for dynamics that underpins almost all further topics in CIE A-Level Physics, from circular motion and energy to particle physics and collision experiments. Mastering the sign conventions and application of conservation of momentum will help you solve a wide range of structured and multiple choice questions across all units. After completing this sub-topic, you can extend your knowledge to more advanced interactions and related topics in dynamics and energy.