P14: Scalar and Vector Quantities
Scalars, vectors and forces
Scalar And Vector Quantities
📋 Key Definitions
Scalar quantity: A physical quantity that has magnitude (size) only. Examples: speed, distance, mass, temperature, energy, time.
Vector quantity: A physical quantity that has both magnitude and direction. Examples: velocity, displacement, force, weight, acceleration, momentum.
Resultant force: A single force that has the same effect as all the individual forces acting on an object combined.
Contact force: A force that acts between two objects that are physically touching. Examples: friction, air resistance, tension, normal contact force.
Non-contact force: A force that acts between two objects that are not physically touching. Examples: gravitational force, magnetic force, electrostatic force.
📐 Scalars vs Vectors
| Property | Scalar | Vector |
|---|---|---|
| Has magnitude? | Yes | Yes |
| Has direction? | No | Yes |
| Examples | Speed, distance, mass, temperature, energy, time | Velocity, displacement, force, weight, acceleration, momentum |
Key pairs to remember: Speed is scalar, velocity is vector. Distance is scalar, displacement is vector. Mass is scalar, weight is vector. The vector versions include direction.
Worked Example 1: Identifying scalars and vectors
Classify each quantity as scalar or vector: (a) 30 m/s, (b) 30 m/s north, (c) 500 J, (d) 10 N downwards.
(a) Scalar — speed has magnitude only, no direction stated
(b) Vector — velocity has magnitude and direction (north)
(c) Scalar — energy has magnitude only
(d) Vector — force has magnitude and direction (downwards)
🤝 Contact and Non-contact Forces
| Force type | Example | Description |
|---|---|---|
| Contact | Friction | Opposes motion between two surfaces in contact |
| Contact | Air resistance | Friction between an object and the air it moves through |
| Contact | Tension | Pulling force in a rope, string or cable |
| Contact | Normal contact force | Support force exerted by a surface on an object resting on it (acts at 90° to surface) |
| Non-contact | Gravity (weight) | Attractive force between masses |
| Non-contact | Magnetic force | Force between magnets or magnetic materials |
| Non-contact | Electrostatic force | Force between charged objects |
Important: Weight is a non-contact force (gravity acts at a distance). Mass is not a force — it is a scalar measure of the amount of matter in an object. Weight = mass × gravitational field strength (W = mg).
➡️ Resultant Force
When multiple forces act on an object, they can be combined into a single resultant force.
Forces in the same direction
Add the magnitudes: if 5 N right and 3 N right act on an object, the resultant force is 8 N right.
Forces in opposite directions
Subtract the smaller from the larger: if 5 N right and 3 N left act on an object, the resultant force is 2 N right.
Worked Example 2: Resultant force in one dimension
Two forces act on a car: 800 N driving force forwards and 300 N drag force backwards. Calculate the resultant force.
Resultant force = 800 − 300 = 500 N forwards
Worked Example 3: Resultant force with multiple forces
Three forces act on an object: 10 N to the right, 4 N to the left, and 2 N to the left. Calculate the resultant force.
Total right = 10 N. Total left = 4 + 2 = 6 N.
Resultant force = 10 − 6 = 4 N to the right
📊 Free Body Diagrams and Resolving Forces
A free body diagram shows all the forces acting on a single object. Each force is drawn as an arrow starting from the object, pointing in the direction the force acts, with length proportional to the force's magnitude.
Resolving forces: A single force at an angle can be split (resolved) into two perpendicular components — usually horizontal and vertical. These components have the same combined effect as the original force.
Worked Example 4: Resolving a force
A force of 50 N acts at 30° above the horizontal. Calculate the horizontal and vertical components.
Horizontal component = 50 × cos 30° = 50 × 0.866 = 43.3 N
Vertical component = 50 × sin 30° = 50 × 0.5 = 25 N
❓ Practice Questions
Q1: Foundation State the difference between a scalar quantity and a vector quantity, giving one example of each.
Q2: Foundation Classify each as scalar or vector: speed, velocity, mass, weight, distance, displacement, energy, force.
Q3: Foundation Give two examples of contact forces and two examples of non-contact forces.
Q4: Higher A boat has a driving force of 600 N forwards and water resistance of 200 N backwards. Calculate the resultant force and state its direction.
Q5: Higher A force of 100 N acts at 40° to the horizontal. Calculate the horizontal and vertical components.
✅ Answers
- A scalar has magnitude only (e.g. speed = 10 m/s). A vector has magnitude and direction (e.g. velocity = 10 m/s north).
- Scalar: speed, mass, distance, energy. Vector: velocity, weight, displacement, force.
- Contact: friction, air resistance, tension, normal contact force. Non-contact: gravity, magnetic force, electrostatic force.
- Resultant force = 600 − 200 = 400 N forwards.
- Horizontal = 100 × cos 40° = 100 × 0.766 = 76.6 N. Vertical = 100 × sin 40° = 100 × 0.643 = 64.3 N.
🎯 Exam Tips
- The easiest way to remember: vector = direction matters. If you need to state which way it points, it's a vector
- Speed/distance are scalars; velocity/displacement are their vector equivalents — don't mix them up
- When calculating resultant force, always state the direction (e.g. “4 N to the right”, not just “4 N”)
- In free body diagrams, only draw forces acting on the object, not forces exerted by the object
- Weight is a force (vector); mass is not a force (scalar) — this is a very common exam trap
- For resolving forces, you only need to use trigonometry at Higher tier
🔢 Maths Skills
Mathematical Skills
For calculating resultant forces, add forces in the same direction and subtract forces in opposite directions. For resolving forces at an angle, use: horizontal component = F × cos θ and vertical component = F × sin θ. Always state the direction of the resultant force.
Maths Example
Forces of 12 N east, 5 N west and 3 N east act on an object. Resultant = (12 + 3) − 5 = 10 N east. A force of 80 N acts at 25° to the horizontal: horizontal = 80 × cos 25° = 80 × 0.906 = 72.5 N; vertical = 80 × sin 25° = 80 × 0.423 = 33.8 N.
⚠️ Common Misconceptions
Watch Out!
1. Wrong: Mass and weight are the same thing. Correct: Mass is the amount of matter (scalar, kg, does not change). Weight is the gravitational force on an object (vector, N, changes with g).
2. Wrong: Newton's 3rd Law force pairs cancel each other out. Correct: The forces act on DIFFERENT objects so they cannot cancel. Only forces on the SAME object can combine to give a resultant force.
3. Wrong: A moving object must have a force acting on it to keep moving. Correct: According to Newton's 1st Law, an object moves at constant velocity with NO resultant force. A force is only needed to change the motion (accelerate/decelerate).
✍️ 6-Mark Question
Extended Answer
6 marks: Explain the difference between scalar and vector quantities. Use examples to illustrate your answer and describe how to calculate the resultant force when multiple forces act on an object.
A scalar quantity has magnitude (size) only, for example speed (30 m/s), distance (50 m) and mass (10 kg). A vector quantity has both magnitude and direction, for example velocity (30 m/s north), displacement (50 m east) and force (10 N downwards). The key difference is that direction matters for vectors but not for scalars. When multiple forces act on an object, they can be combined into a single resultant force. If forces act along the same line in the same direction, they are added together. If they act in opposite directions, the smaller force is subtracted from the larger one and the resultant is in the direction of the larger force. For forces at an angle, the forces can be resolved into perpendicular components using trigonometry, and the horizontal and vertical components can then be combined separately. The resultant force determines the acceleration of the object according to F = ma.
Mark scheme: 1 mark for scalar = magnitude only with example; 1 mark for vector = magnitude + direction with example; 1 mark for key difference (direction); 1 mark for adding forces in same direction; 1 mark for subtracting opposite forces and stating direction; 1 mark for resolving at angles or linking to F = ma. (6 marks total)
📊 AO3: Analyse & Evaluate
Analysis and Evaluation
A student draws a free body diagram for a book resting on a table. They show two forces: weight (10 N downwards) and normal contact force (10 N upwards). Another student says these are a Newton's 3rd Law force pair.
(a) Explain why the two forces are NOT a Newton's 3rd Law pair, even though they are equal and opposite.
(b) State the actual Newton's 3rd Law pair for the weight of the book.
(c) A third student says the book does not fall because the normal force "cancels" the weight. Evaluate this explanation.
Answers: (a) Newton's 3rd Law pairs must act on DIFFERENT objects and be the SAME TYPE of force. Weight and normal contact force both act on the book (same object) and are different types (gravitational vs contact), so they are not a 3rd Law pair. (b) The Earth pulls the book down (weight) and the book pulls the Earth up (gravitational force on Earth). Both are gravitational forces acting on different objects. (c) The explanation is partly correct but poorly worded. The forces do balance (resultant force = 0), which is why the book does not accelerate (Newton's 1st Law). However, "cancels" is misleading — both forces still exist. The correct explanation is: the resultant force is zero, so the book remains stationary (Newton's 1st Law), not that the forces "cancel out of existence".
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