OCR Specification focus:
‘Define linear momentum p = mv; treat momentum as a vector in calculations and diagrams.’
Momentum is a central concept in mechanics, linking mass and velocity to describe the motion of objects. Understanding momentum as a vector quantity is vital in analysing interactions and motion changes.
Momentum as a Physical Quantity
Momentum describes how difficult it is to stop or change the motion of an object. It combines both mass and velocity into one measurable property. Since velocity has direction, momentum also possesses direction, making it a vector. This means it must be represented using arrows or components in vector diagrams, especially when dealing with two-dimensional motion.
Definition of Linear Momentum
Linear Momentum (p): The product of an object’s mass and its velocity. It is a vector quantity pointing in the same direction as the velocity.
An object with a large mass moving slowly can have the same momentum as a light object moving quickly — both mass and velocity are equally important contributors. Momentum quantifies the quantity of motion, enabling comparisons across varied systems.
The Equation for Momentum
EQUATION
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Momentum (p) = m × v
m = mass (kilograms, kg)
v = velocity (metres per second, m s⁻¹)
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The SI unit of momentum is the kilogram metre per second (kg m s⁻¹). In practical problems, momentum helps predict the effects of collisions, recoil, and motion transfer in systems from subatomic particles to planets.
Understanding Momentum as a Vector
Momentum is not just about magnitude; it also has direction. The direction of momentum is identical to that of the object’s velocity vector.
This figure shows a football (soccer) with velocity and momentum arrows pointing in the same direction, emphasising that momentum inherits the direction of velocity. The visual focuses on direction only, consistent with treating momentum as a vector. No additional concepts beyond the syllabus focus are included. Source
In calculations:
When two or more momenta act in different directions, vector addition or subtraction is required.
A clear vector-addition diagram showing standard head-to-tail and parallelogram constructions for finding the resultant. Use the same constructions when combining multiple momentum vectors. The image contains only essential geometry and no extra material beyond the sub-subtopic. Source
Opposite directions must be treated as having opposite signs (e.g. positive to the right, negative to the left).
In two-dimensional systems, momentum is often split into horizontal (x) and vertical (y) components using trigonometric methods.
A generic vector-components diagram suitable for momentum p\mathbf{p}p illustrates decomposition into x and y components. This aligns with the required treatment of momentum as a vector in two dimensions. The diagram is minimal and avoids extraneous information. Source
This vector nature means scalar arithmetic cannot be used. Instead, vector diagrams and component analysis ensure accurate results, particularly in collision and explosion contexts where direction strongly influences the outcome.
Representing Momentum with Diagrams
Momentum vectors are often drawn as arrows scaled to magnitude and oriented to indicate direction. These diagrams help visualise:
How momenta combine or cancel.
The net momentum of a system, found by vector addition of individual momenta.
The effects of external forces acting in various directions.
In one-dimensional cases, directions can be simplified using positive and negative signs. In two dimensions, the Pythagorean theorem and trigonometric functions (sine, cosine, tangent) are used to find resultant magnitudes and directions.
Relation Between Force and Momentum
Newton’s Second Law connects the net external force acting on an object to the rate of change of momentum. This concept is formally introduced in a later subsubtopic, but it is useful here to note the direct link between momentum and force.
When a force acts on a body, it changes its velocity and therefore its momentum. The direction of the change in momentum is determined by the direction of the net force. This connection underlines why momentum is a key descriptor of motion: it changes only when an external influence acts.
Scalar versus Vector Treatment
It is essential to distinguish between scalar and vector treatments of physical quantities:
Mass is a scalar, having magnitude only.
Velocity is a vector, possessing both magnitude and direction.
Momentum, therefore, inherits the vector properties of velocity.
This means:
Reversing direction reverses the sign of momentum.
Equal magnitudes of mass and velocity moving in opposite directions result in momenta that cancel when combined.
Failure to include direction can produce incorrect conclusions about system motion, especially in conservation problems.
The Importance of Momentum in System Analysis
Momentum provides a framework for understanding interactions between objects. In an isolated system (one without external forces), the total momentum remains constant. While this principle formally belongs to later sections of the specification, recognising momentum as a vector quantity prepares students to apply conservation laws accurately in both one- and two-dimensional scenarios.
Momentum vectors allow:
The tracking of motion changes before and after collisions.
The prediction of recoil and separation behaviour.
The identification of resultant system motion when multiple bodies interact.
In practical contexts, physicists use this understanding to study collisions in car safety analysis, recoil in firearms, or momentum transfer in particle physics.
Vector Components in Momentum Calculations
To analyse momentum in two dimensions:
Resolve each momentum vector into x and y components.
Apply conservation or force laws separately to each component.
Recombine the components to find resultant momentum or velocity using vector addition.
This approach ensures accurate directional outcomes, essential in both theoretical and experimental physics.
Common Pitfalls in Momentum Analysis
Students often make errors by:
Ignoring vector direction and using scalar arithmetic.
Forgetting to assign positive or negative signs to opposing directions.
Mixing up mass and weight (weight is a force, not a property of motion).
Failing to maintain consistent units when calculating p = mv.
Consistent use of vector notation, unit analysis, and clear diagrammatic reasoning helps avoid these mistakes.
Summary of Key Ideas
Although a full summary is not required here, students should note the critical learning outcomes embedded within this subsubtopic:
Momentum is defined as p = mv and is a vector quantity.
It depends on both mass and velocity, carrying direction identical to the object’s motion.
Correct treatment requires vector addition, component resolution, and attention to sign conventions.
Momentum, therefore, serves as a foundational bridge between Newton’s laws of motion and later principles such as impulse, force-time relationships, and momentum conservation in collisions and interactions.
FAQ
In one dimension, the conservation of momentum is applied along a straight line, so direction can be represented using positive and negative signs.
In two dimensions, momentum must be conserved independently in both x and y directions. This means:
Resolve each momentum vector into components.
Apply conservation separately to each component.
Recombine components using vector addition to find the resultant momentum or velocity.
Momentum combines both mass and velocity, giving an indication of how difficult it is to stop an object.
A larger mass or higher velocity both increase momentum, meaning the object carries more “motion content.”
This idea helps explain why a slow-moving lorry can have more momentum than a fast-moving tennis ball — it depends on both size and speed together.
Momentum vectors are shown as arrows where:
The length represents the magnitude (how large the momentum is).
The direction shows the motion’s orientation.
In 2D problems, multiple momentum vectors can be combined using:
Head-to-tail diagrams for sequential addition.
Parallelogram constructions for simultaneous vector addition.
These graphical tools help visualise resultant momentum and ensure direction is properly treated.
Yes. Momentum depends on mass and velocity, while kinetic energy depends on velocity squared.
For example:
A heavy object moving slowly can have the same momentum as a light object moving quickly.
However, because kinetic energy depends on velocity squared, the lighter, faster object will have greater kinetic energy even if both share the same momentum.
This distinction shows that momentum and kinetic energy are related but not directly proportional.
Momentum’s vector nature can be demonstrated in laboratory settings such as:
Air track experiments, where gliders collide on a frictionless surface, allowing observation of direction-dependent momentum changes.
Two-dimensional collision setups, using pucks on air tables or low-friction surfaces to visualise vector addition and conservation.
By measuring mass, velocity, and direction before and after interaction, students can confirm that both magnitude and direction must be included for accurate momentum analysis.
Practice Questions
Question 1 (2 marks)
Define linear momentum and state whether it is a scalar or vector quantity, explaining why.
Mark Scheme:
1 mark for correct definition: Momentum is the product of an object’s mass and its velocity (p = mv).
1 mark for correct identification and explanation: Momentum is a vector quantity because it has both magnitude and direction, the same as velocity.
Question 2 (5 marks)
A 1.5 kg trolley moves east at 4.0 m s⁻¹. It collides with and sticks to a stationary 0.5 kg trolley.
(a) Calculate the momentum of the 1.5 kg trolley before the collision.
(b) Determine the velocity (magnitude and direction) of the combined trolleys immediately after the collision.
(c) Explain why momentum, treated as a vector, must be conserved in this situation.
Mark Scheme:
(a) 1 mark for correct substitution into p = mv: p = 1.5 × 4.0 = 6.0 kg m s⁻¹.
(b) 1 mark for correct use of total momentum before = total momentum after.
(b) 1 mark for correct calculation of total mass (1.5 + 0.5 = 2.0 kg).
(b) 1 mark for correct velocity after collision: v = 6.0 / 2.0 = 3.0 m s⁻¹ east.
(c) 1 mark for clear statement that momentum is conserved in a closed system with no external forces, and direction must be included because momentum is a vector quantity.
