Motion
Speed and Velocity
Speed tells you how fast an object is moving. It is a scalar quantity — it has magnitude but no direction. The equation linking distance, speed and time is:
$$s = v \times t$$
Where \(s\) is distance (m), \(v\) is speed (m/s), and \(t\) is time (s). You can rearrange this to find speed (\(v = \frac{s}{t}\)) or time (\(t = \frac{s}{v}\)). Typical speeds to know include: walking (~1.5 m/s), running (~3 m/s), cycling (~6 m/s), a car in town (~13 m/s or 30 mph), and the speed of sound in air (~340 m/s).
Velocity is speed in a stated direction. It is a vector quantity. An object travelling in a circle at constant speed has a changing velocity because its direction constantly changes — meaning it is always accelerating, even though its speed stays the same.
Similarly, distance (scalar) measures total ground covered, while displacement (vector) measures how far and in what direction from the start. A runner who completes a 400 m lap of a track covers a distance of 400 m but has a displacement of zero — they end up back where they started.
Speed and distance are scalars (magnitude only). Velocity and displacement are vectors (magnitude + direction). An object moving in a circle at constant speed has a changing velocity.
Acceleration
Acceleration is the rate at which velocity changes. An object accelerates when it speeds up, slows down, or changes direction. The equation is:
$$a = \frac{v - u}{t}$$
Where \(a\) is acceleration (\(\text{m/s}^2\)), \(v\) is final velocity (m/s), \(u\) is initial velocity (m/s), and \(t\) is time (s). A negative value of acceleration means the object is decelerating (slowing down). For example, a car braking from 20 m/s to 0 m/s in 5 s has an acceleration of \(a = \frac{0 - 20}{5} = -4 \text{ m/s}^2\).
For uniform (constant) acceleration, you also need the equation:
$$v^2 = u^2 + 2as$$
Where \(s\) is distance (m). This is useful when time is not given in the question. It is particularly important for braking distance calculations.
Question: A car accelerates from 5 m/s to 25 m/s in 10 seconds. Calculate its acceleration.
Answer: \(a = \frac{v - u}{t} = \frac{25 - 5}{10} = \frac{20}{10} = 2 \text{ m/s}^2\)
Distance-Time Graphs
On a distance-time graph, the gradient (slope) of the line equals the speed. You can interpret the shape of the line as follows:
• A horizontal line means the object is stationary (speed = 0).
• A straight diagonal line means constant speed — the steeper the gradient, the faster the speed.
• A curve getting steeper means the object is accelerating.
• A curve getting shallower means the object is decelerating.
To find the speed at a specific instant on a curved d-t graph, draw a tangent to the curve at that point and calculate the gradient of the tangent.
Velocity-Time Graphs
On a velocity-time graph, two key pieces of information can be extracted:
• The gradient equals the acceleration. A steeper slope means greater acceleration. A horizontal line means constant velocity (zero acceleration). A downward slope means deceleration.
• A line below the x-axis means the object is moving in the opposite direction.
The area under the line on a velocity-time graph equals the distance travelled. For a triangle, the area is \(\frac{1}{2} \times \text{base} \times \text{height}\). For a rectangle, it is \(\text{base} \times \text{height}\). For irregular shapes, split the area into triangles and rectangles and add them together.
On a distance-time graph, gradient = speed. On a velocity-time graph, gradient = acceleration.
On a velocity-time graph, the area under the line = distance travelled. Split complex shapes into triangles and rectangles to calculate the total area.
Any moving object has kinetic energy. The equation is:
$$E_k = \frac{1}{2} \times m \times v^2$$
Where \(E_k\) is kinetic energy (J), \(m\) is mass (kg) and \(v\) is speed (m/s). Because speed is squared, doubling the speed quadruples the kinetic energy. This has critical implications for braking distances and road safety: a car travelling at 40 mph has four times the kinetic energy of one at 20 mph, meaning it needs four times the braking distance to stop.
Average Speed and Non-Uniform Motion
Most real journeys involve varying speeds. When an object's speed varies, you calculate average speed by dividing the total distance by the total time:
$$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$
On a distance-time graph with a curved line, you can estimate speed at a particular moment by drawing a tangent and finding its gradient. The average speed between two points is found by drawing a straight line (chord) between them and calculating its gradient.
To measure the speed of a trolley rolling down a ramp, mark a measured distance on the ramp with two markers. Use a stopwatch to time how long the trolley takes to travel between them, then calculate speed using \(v = \frac{s}{t}\). For greater precision, use light gates connected to a data logger — these measure very short time intervals as the trolley passes through, allowing you to calculate instantaneous speed. To investigate acceleration, use two light gates at different positions and compare the speeds.