Topic 1.6: Momentum Revision Notes

1. Overview

Momentum is a fundamental concept in physics that describes the "quantity of motion" an object possesses. It is a vector quantity that accounts for both the mass and the velocity of an object, helping us predict the outcomes of collisions and explosions in everyday life, from car crashes to sports.

Key Definitions

Momentum: The product of an object's mass and its velocity.
Impulse: The product of the force acting on an object and the time for which it acts, resulting in a change in momentum.
Conservation of Momentum: The principle stating that the total momentum of a closed system remains constant, provided no external forces act on it.
Resultant Force: In terms of momentum, it is defined as the rate of change of momentum per unit time.

Core Content

Note: The IGCSE syllabus classifies the specific learning objectives for Momentum under the Supplement (Extended) curriculum. Refer to the section below for full details.

Extended Content (Extended curriculum only)

A. Calculating Momentum

Momentum ($p$) depends on mass and velocity. Because velocity is a vector, momentum is also a vector. This means the direction is just as important as the magnitude.

Equation: $p = mv$
If an object is at rest (velocity = 0), its momentum is 0.

B. Impulse and Change in Momentum

Impulse describes the effect of a force acting over a period of time. When a resultant force acts on an object, it causes the object's velocity to change, which means its momentum changes.

Equation: $Impulse = F\Delta t = \Delta(mv)$
$\Delta(mv)$ represents the change in momentum (Final Momentum – Initial Momentum).

📊A tennis racket hitting a ball. An arrow shows the force (F) applied over a short time interval (t), with the ball's velocity changing from 'u' (approaching) to 'v' (moving away).

C. The Conservation of Momentum

In any collision or explosion involving two or more objects, the total momentum before the event is equal to the total momentum after the event, provided no external forces (like friction) act.

Total momentum before = Total momentum after
$(m_1 \times u_1) + (m_2 \times u_2) = (m_1 \times v_1) + (m_2 \times v_2)$

Worked Example: A 2 kg trolley moving at 3 m/s hits a stationary 1 kg trolley. They stick together and move off. Calculate their common velocity.

Momentum before = $(2 \text{ kg} \times 3 \text{ m/s}) + (1 \text{ kg} \times 0 \text{ m/s}) = 6 \text{ kg m/s}$.
Momentum after = $(2 \text{ kg} + 1 \text{ kg}) \times v = 3v$.
Apply conservation: $6 = 3v$.
Solve: $v = 2 \text{ m/s}$.

D. Force as Rate of Change of Momentum

Newton's Second Law can be expressed more precisely using momentum. The resultant force acting on an object is equal to the change in momentum divided by the time taken for that change.

Equation: $F = \frac{\Delta p}{t}$ or $F = \frac{mv - mu}{t}$

Key Equations

Equation	Symbols	Units
$p = mv$	$p$ = momentum, $m$ = mass, $v$ = velocity	kg m/s, kg, m/s
$Impulse = F\Delta t$	$F$ = Force, $\Delta t$ = time interval	N s (Newton-seconds)
$\Delta p = mv - mu$	$mu$ = initial momentum, $mv$ = final momentum	kg m/s
$F = \frac{\Delta p}{t}$	$F$ = Resultant Force, $t$ = time	N, kg m/s, s

Common Mistakes to Avoid

❌ Wrong: Calculating change in momentum for a rebounding ball by subtracting speeds (e.g., $10 \text{ m/s} - 7 \text{ m/s} = 3 \text{ m/s}$).
- ✅ Right: Treat velocity as a vector. If a ball hits a wall at $+10 \text{ m/s}$ and bounces back at $7 \text{ m/s}$, its new velocity is $-7 \text{ m/s}$. The change in velocity is $10 - (-7) = 17 \text{ m/s}$.
❌ Wrong: Dividing mass by velocity ($m/v$) to find momentum.
- ✅ Right: Always multiply mass and velocity ($mv$).
❌ Wrong: Forgetting to convert time into seconds (e.g., using milliseconds or minutes directly in the $F = \Delta p/t$ formula).
- ✅ Right: Always ensure time is in seconds (s) and mass is in kilograms (kg).
❌ Wrong: Assigning a positive sign to momentum in both directions in a collision.
- ✅ Right: Assign one direction as positive (e.g., Right = $+$) and the opposite as negative (e.g., Left = $-$).

Exam Tips

Check for "Sticking Together": In collision questions, check if the objects move off separately or stick together. If they stick together, add their masses for the "momentum after" calculation.
The Unit of Impulse: Remember that Impulse can be measured in either $N \cdot s$ or $kg \cdot m/s$. They are equivalent!
Show Your Workings: In momentum conservation problems, clearly state "Total momentum before = Total momentum after" to pick up easy method marks, even if you make a calculation error later.