2.1.1 Scalar and vector quantities
Scalar and vector quantities
Definition
Scalar quantity
A scalar quantity has magnitude but no specific direction, such as distance, speed, mass or energy.
Definition
Vector quantity
A vector quantity has both magnitude and a specific direction, such as displacement, velocity, acceleration or force.
- A physical quantity needs a magnitude and a unit. Magnitude means its numerical size.
- A scalar is completely described by its magnitude and unit, such as a mass of 70 kg70\ \text{kg}70 kg or an energy transfer of 84 kJ84\ \text{kJ}84 kJ.
- A vector also needs a direction, such as a force of 5 N5\ \text{N}5 N downwards or a velocity of 12 m/s12\ \text{m/s}12 m/s east.
- A vector can be represented by an arrow. The arrow points in the vector’s direction, while its length can be drawn to scale to represent the magnitude.
Classifying quantities
- The scalar quantities are distance, speed, mass and energy.
- The vector quantities are displacement, velocity, acceleration, force, weight and momentum.
- Distance is the total length of the path travelled, measured in m\text{m}m.
- Displacement is the straight-line change in position from the starting point to the finishing point, stated with a direction and measured in m\text{m}m.
- Speed describes how quickly distance is travelled and is measured in m/s\text{m/s}m/s.
- Velocity is speed in a stated direction and is measured in m/s\text{m/s}m/s.
- Mass measures the amount of matter in an object and is measured in kg\text{kg}kg.
- Weight is the gravitational force acting on an object, measured in N\text{N}N and directed towards the centre of the attracting body.
- Acceleration, force and momentum are vectors because changing their direction changes the physical situation. Energy is a scalar because it has no direction.
Common Mistake
- Do not use mass and weight as interchangeable terms. Mass is a scalar measured in kg\text{kg}kg, while weight is a vector force measured in N\text{N}N.
- Do not identify a vector only from its unit. Distance and displacement both use m\text{m}m, while speed and velocity both use m/s\text{m/s}m/s.
Direction changes vector results
- Choose one direction as positive when motion is along a straight line. Quantities in the opposite direction are negative.
- A velocity of −12 m/s-12\ \text{m/s}−12 m/s means a velocity of magnitude 12 m/s12\ \text{m/s}12 m/s in the chosen negative direction. It does not mean a speed below zero.
- Vectors in opposite directions can partly or completely cancel. Scalars do not cancel because they carry no direction.
Example
- A walker travels 30 m30\ \text{m}30 m east and then 10 m10\ \text{m}10 m west.
- The distance is 30+10=40 m30+10=40\ \text{m}30+10=40 m because distance counts the whole route.
- Taking east as positive, the displacement is 30−10=20 m30-10=20\ \text{m}30−10=20 m east.
Speed and velocity
Definition
Velocity
Velocity is speed in a stated direction, measured in metres per second.
- An object can have constant speed while its velocity changes. When it moves around a curve, the direction changes even if the magnitude of the velocity stays constant.
- After one complete lap of a circular track, the distance is the circumference of the track but the displacement is zero because the finishing position is the starting position.
Example
- A runner completes one 400 m400\ \text{m}400 m lap in 50 s50\ \text{s}50 s.
- The average speed is 40050=8.0 m/s\dfrac{400}{50}=8.0\ \text{m/s}50400=8.0 m/s.
- The displacement is 0 m0\ \text{m}0 m, so the average velocity is 050=0 m/s\dfrac{0}{50}=0\ \text{m/s}500=0 m/s.
Exam technique
- When asked to compare a scalar with a vector, state that both have magnitude and then add that only the vector has a specific direction.
- For a displacement or velocity answer, include the direction. A magnitude and unit alone give only distance or speed.
- Use signs consistently in one-dimensional calculations and state which direction you selected as positive.
Self review
- What information describes a scalar quantity completely?
- What additional information is required for a vector quantity?
- Which listed motion quantities form the scalar and vector pairs?
- Why can velocity change while speed remains constant?
