This note covers speed, velocity and acceleration, and how to read distance-time and speed-time graphs. These graphs turn a story about movement into numbers: the gradient and the area under a graph each have a physical meaning you must be able to use.
Key ideas
Speed is how fast an object moves, regardless of direction. Velocity is speed in a stated direction, so it is a vector. Two cars at 20 m/s travelling in opposite directions have the same speed but different velocities.
For a journey with changing speed, use average speed = total distance ÷ total time. The speed shown on a car speedometer is the speed at one instant.
Acceleration is the rate of change of velocity. If velocity increases, acceleration is positive. If velocity decreases, acceleration is negative, often called deceleration. Acceleration is also a vector.
Distance-time graphs
- A horizontal line means the object is stationary.
- A straight sloping line means constant speed.
- A curve getting steeper means the object is speeding up; a curve getting flatter means it is slowing down.
- The gradient (rise ÷ run) equals the speed. For a curve, draw a tangent at the point and find its gradient.
Speed-time graphs
- A horizontal line means constant speed (zero acceleration).
- A straight sloping line means constant acceleration. Upward slope is speeding up; downward slope is slowing down.
- The gradient equals the acceleration.
- The area under the graph equals the distance travelled. Split the area into rectangles and triangles, or count squares for a curve.
Free fall
Near the Earth's surface, an object falling with negligible air resistance accelerates downwards at about 9.8 m/s2. This acceleration is the same for all masses. In air, resistance grows with speed, so the acceleration falls until the object reaches a steady (terminal) speed. That is covered in the note on forces.
Curved graphs and non-uniform motion (Extended)
Real motion is rarely made of perfect straight lines. On a speed-time graph, a curve that gets steeper means the acceleration is increasing, and a curve that flattens towards a horizontal line means the acceleration is decreasing towards zero. A falling object with air resistance gives exactly this kind of curve. To find the acceleration at one moment, draw a tangent to the curve at that point and calculate its gradient. To find the distance for a curved graph, count the squares under the line and multiply the number by the value of one square (width in s multiplied by height in m/s).
Describing a journey from a graph
A good description names each section in turn and gives numbers. For example: "From 0 to 8 s the car accelerates uniformly at 2.5 m/s2. From 8 s to 23 s it moves at a constant 20 m/s. From 23 s to 28 s it decelerates uniformly to rest." Always say what the line is doing, what that means for the motion, and support it with a value read from the axes.
Speed, velocity and direction (Extended: circular motion)
An object moving at constant speed around a circle keeps changing direction, so its velocity is changing even though its speed is not. That means it is accelerating. Acceleration therefore means a change in speed, a change in direction, or both. For straight-line motion, a negative value simply means the object is slowing down or travelling in the opposite direction to the one chosen as positive.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| speed = distance ÷ time | Units m/s. Use average speed = total distance ÷ total time for a whole journey. |
| Velocity | Speed with direction (vector). |
| a = (v − u) ÷ t | Acceleration from change in speed (v final, u initial) over time t. Units m/s2. |
| Gradient of distance-time graph | Speed. |
| Gradient of speed-time graph | Acceleration. |
| Area under speed-time graph | Distance travelled. |
| Free-fall acceleration, g | About 9.8 m/s2 near the Earth's surface. |
| 1 km/h | Equals 1000 ÷ 3600 = 0.278 m/s. |
Worked examples
Example 1 A train travels 150 km in 1.5 hours. Find its average speed in km/h and in m/s.
Step 1: average speed = 150 ÷ 1.5 = 100 km/h.
Step 2: convert: 100 × 1000 ÷ 3600 = 27.8 m/s.
Answer: 100 km/h, which is 27.8 m/s.
Example 2 A car's speed-time graph is made of three straight sections. It accelerates uniformly from rest to 20 m/s in 8.0 s, then travels at 20 m/s for 15 s, then slows uniformly to rest in 5.0 s. Find (a) the acceleration in the first section, (b) the deceleration in the last section, (c) the total distance, (d) the average speed.
(a) a = (20 − 0) ÷ 8.0 = 2.5 m/s2.
(b) a = (0 − 20) ÷ 5.0 = −4.0 m/s2, so the deceleration is 4.0 m/s2.
(c) Area of triangle 1 = ½ × 8.0 × 20 = 80 m. Area of rectangle = 20 × 15 = 300 m. Area of triangle 2 = ½ × 5.0 × 20 = 50 m. Total = 80 + 300 + 50 = 430 m.
(d) Total time = 8.0 + 15 + 5.0 = 28 s. Average speed = 430 ÷ 28 = 15.4 m/s.
Answer: (a) 2.5 m/s2; (b) 4.0 m/s2 deceleration; (c) 430 m; (d) 15.4 m/s.
Example 3 A stone is dropped from a bridge. Ignoring air resistance, find its speed after 3.0 s and the distance it has fallen. Use g = 9.8 m/s2.
Step 1: it starts from rest, so u = 0. From a = (v − u) ÷ t, v = a × t = 9.8 × 3.0 = 29.4 m/s.
Step 2: the speed-time graph is a straight line from 0 to 29.4 m/s over 3.0 s. The distance is the area of a triangle: ½ × 3.0 × 29.4 = 44.1 m.
Answer: 29.4 m/s and 44.1 m.
Common errors
- Reading the gradient of a distance-time graph as acceleration. It is speed.
- Reading the height of a speed-time graph as distance. Distance is the area underneath.
- Describing a horizontal line on a distance-time graph as "constant speed". It means the object is at rest.
- Averaging speed by adding two speeds and halving, when the times for each speed were different. Always use total distance ÷ total time.
- Forgetting to convert km/h or minutes to m/s and s before using an equation.
Exam tip: When asked to find a gradient or an area, draw the triangle or rectangle on the graph, show the numbers you read from the axes, and include the unit of the final answer.
Check yourself
- A cyclist rides 9.0 km in 30 minutes. What is the average speed in m/s?
Answer
Distance = 9000 m, time = 1800 s. Speed = 9000 ÷ 1800 = 5.0 m/s. - A sprinter reaches 10 m/s from rest in 2.5 s. What is the acceleration?
Answer
a = (10 − 0) ÷ 2.5 = 4.0 m/s2. - A speed-time graph shows a constant 6.0 m/s for 10 s. How far has the object travelled?
Answer
Area of rectangle = 6.0 × 10 = 60 m. - What does a straight, steeply sloping line on a distance-time graph tell you compared with a gently sloping line?
Answer
A steeper line has a larger gradient, so the object moves at a greater constant speed.