Scalars, vectors and speed

Scalar quantities have magnitude only. Vector quantities have magnitude and an associated direction.

A vector quantity may be represented by an arrow. The length of the arrow represents the magnitude, and the direction of the arrow the direction of the vector quantity.

Both parts of that convention are marked: an arrow drawn twice as long must represent twice the force, and the arrow must point the way the force acts.

This first part of the sub-topic is about describing motion — how far, how fast, and how quickly the speed changes — before any forces are brought in.

Four quantities do the work, and they come in two pairs. Distance and displacement both say how far; speed and velocity both say how fast. In each pair the second one includes a direction and the first does not, and that single difference is what the first three sections below are about.

Throughout this section on forces and motion you should be able to use ratios and proportional reasoning to convert units and to compute rates.

Distance is how far an object moves. Distance does not involve direction. Distance is a scalar quantity.

Displacement includes both the distance an object moves, measured in a straight line from the start point to the finish point, and the direction of that straight line. Displacement is a vector quantity.

You should be able to express a displacement in terms of both the magnitude and direction.

The difference shows up whenever a path is not straight. Walk 3 km north then 4 km east and the distance travelled is 7 km, but the displacement is 5 km in a north-easterly direction — the straight line from start to finish. Walk a complete lap of a track and the distance is a lap, but the displacement is zero.

Speed does not involve direction. Speed is a scalar quantity.

The speed of a moving object is rarely constant. When people walk, run or travel in a car their speed is constantly changing. The speed at which a person can walk, run or cycle depends on many factors, including age, terrain, fitness and distance travelled.

You should be able to recall typical values of speed, so these are worth learning as numbers.

You should also be able to recall typical values of speed for different types of transportation systems — a car on a motorway is roughly 30 m/s, a train perhaps 55 m/s, and a passenger aeroplane around 250 m/s.

For an object moving at constant speed, the distance travelled in a specific time can be calculated using the equation:

distance travelled = speed × time s = v t

distance, s, in metres, m; speed, v, in metres per second, m/s; time, t, in seconds, s.

You should be able to recall and apply this equation, to make measurements of distance and time and then calculate speeds of objects, and to calculate average speed for non-uniform motion.

The velocity of an object is its speed in a given direction. Velocity is a vector quantity.

You should be able to explain the vector–scalar distinction as it applies to displacement, distance, velocity and speed. Distance and speed are scalars, with magnitude only; displacement and velocity are vectors, with a direction as well.

If an object moves along a straight line, the distance travelled can be represented by a distance–time graph.

The speed of an object can be calculated from the gradient of its distance–time graph.

You should be able to draw distance–time graphs from measurements, extract and interpret lines and slopes of distance–time graphs, translating information between graphical and numerical form, and determine speed from a distance–time graph.

Higher tier: if an object is accelerating, its speed at any particular time can be determined by drawing a tangent and measuring the gradient of the distance–time graph at that time.

Draw the tangent so that it just touches the curve at the time asked for, extend it well across the graph, and take a large triangle from the tangent itself.