Motion: Speed, Velocity and Acceleration
Scalars and Vectors
Every physical quantity is either a scalar or a vector. Scalars have size only. Vectors have both size and direction. This distinction matters in physics: a car travelling at 20 m/s northwards and another travelling at 20 m/s southwards have the same speed but very different velocities.
Distance is a scalar — it measures the total length of path covered, with no regard for direction. Displacement is a vector — it measures how far an object is from its starting point in a straight line, and in which direction. A runner completing one lap of a 400 m track has covered 400 m distance but zero displacement.
Speed and Velocity
Speed is a scalar — it tells you how fast an object is moving but not the direction. The equation for speed is:
$$v = \frac{s}{t}$$
Where \(v\) is speed in m/s, \(s\) is distance in metres, and \(t\) is time in seconds. Rearranging gives \(s = vt\) and \(t = s / v\).
Typical speeds: walking ~1.5 m/s; running ~3 m/s; cycling ~6 m/s; car in a town ~13 m/s; car on a motorway ~30 m/s; sound in air ~330 m/s. These values are given as context — know the order of magnitude.
Velocity is a vector. It has both magnitude (speed) and direction. An object moving in a circle at constant speed has a changing velocity because its direction changes continuously — this means it is accelerating even though its speed is constant.
Acceleration
Acceleration measures how quickly velocity changes. The equation is:
$$a = \frac{v - u}{t}$$
Where \(a\) is acceleration in m/s2, \(v\) is final velocity in m/s, \(u\) is initial velocity in m/s, and \(t\) is time in seconds. A negative value for \(a\) means the object is decelerating (slowing down).
For uniform (constant) acceleration when time is not given: \(v^2 = u^2 + 2as\), where \(s\) is the distance covered. This equation is provided on the equation sheet — but you must be able to rearrange and apply it correctly.
A train accelerates uniformly from rest and travels 400 m before reaching 20 m/s. Calculate its acceleration.
Given: \(u = 0\) m/s, \(v = 20\) m/s, \(s = 400\) m. Using \(v^2 = u^2 + 2as\):
\(20^2 = 0^2 + 2 \times a \times 400\)
\(400 = 800a\)
\(a = \frac{400}{800} = 0.5\) m/s2
Uniform vs Non-Uniform Acceleration
When an object experiences a constant resultant force, its acceleration is uniform — the same value throughout. A car on a straight motorway with constant engine force and negligible air resistance would have uniform acceleration. In contrast, real vehicles and falling objects experience non-uniform acceleration because resistive forces (air resistance, drag) increase with speed, gradually reducing the resultant force and therefore the acceleration.
On a velocity–time graph, uniform acceleration appears as a straight line (constant gradient). Non-uniform acceleration appears as a curve where the gradient decreases over time — such as the velocity of a sky diver before reaching terminal velocity, where the line levels off as drag builds up.
Distance–Time Graphs
A distance–time graph shows how far an object has moved from a reference point over time. The gradient (slope) of the line gives the speed.
- A horizontal line means the object is stationary (gradient = 0, speed = 0).
- A straight diagonal line means constant speed (constant gradient).
- A curve getting steeper means increasing speed (acceleration). The instantaneous speed at any point on a curve is found by drawing a tangent at that point and calculating its gradient.
Velocity–Time Graphs
A velocity–time graph shows how velocity changes over time. Two key rules:
- Gradient = acceleration. A steeply rising line means large acceleration; a falling line means deceleration.
- Area under the graph = displacement (distance). Split the area into rectangles and triangles to calculate the total distance travelled.
A car accelerates from 0 to 20 m/s in 10 s (triangle shape on the graph), then travels at 20 m/s for a further 20 s (rectangle). Calculate the total distance travelled.
Triangle area = ½ × base × height = ½ × 10 × 20 = 100 m
Rectangle area = 20 × 20 = 400 m
Total distance = 100 + 400 = 500 m
Investigating Motion: Core Practical
A core practical for this topic involves investigating the relationship between acceleration and force or mass, often using a dynamics trolley and light gates on a measured track. Students measure the time for the trolley to pass through each light gate, calculate the change in velocity, and determine acceleration. By varying the applied force (using hanging masses on a pulley string) while keeping the trolley mass constant, the effect of force on acceleration can be plotted as a graph.
The key practical skills assessed include: correctly using light gates to measure velocity, identifying the independent variable (force or mass), measuring the dependent variable (acceleration), and recognising sources of error such as friction on the runway. Compensating for friction by tilting the runway slightly is a common technique — the trolley should move at constant speed without the hanging mass before adding a force.