Reading Motion Graphs

A bus leaves the stop, speeds up, travels steadily and then brakes. You could describe this in words. A graph shows it in one picture. This note helps you read that picture.

The two graphs and what to look at

Distance-time graph (distance on the up axis, time along the bottom):

• The slope is the speed. A steeper line means a faster object.
• A flat line means the object is at rest. The distance does not change.
• A straight sloping line means a steady speed.
• A line that curves upward means the object is speeding up.

Velocity-time graph (velocity on the up axis, time along the bottom):

• The slope is the acceleration: $a = \dfrac{\text{change in velocity}}{\text{time taken}}$.
• A flat line means a steady velocity. The acceleration is zero.
• A line sloping up means speeding up. A line sloping down means slowing down.
• The area under the line is the distance travelled.

See it in numbers

A car starts from rest. It speeds up to 20 m/s in 10 s. It then keeps 20 m/s for 20 s. Then it brakes to rest in the next 10 s.

• First part: $a = \dfrac{20 - 0}{10} = 2$ m/s². The area is a triangle: $\tfrac{1}{2} \times 10 \times 20 = 100$ m.
• Second part: the line is flat, so $a = 0$. The area is a rectangle: $20 \times 20 = 400$ m.
• Third part: $a = \dfrac{0 - 20}{10} = -2$ m/s². The area is a triangle: $\tfrac{1}{2} \times 10 \times 20 = 100$ m.
• Total distance: $100 + 400 + 100 = 600$ m. Average speed: $\dfrac{600}{40} = 15$ m/s.

Run this program to draw the graph. Then look at the three parts.

Mistakes to avoid

• Reading the height of a velocity-time graph as the distance. The height is the velocity. The distance is the area.
• Thinking that a flat line on a distance-time graph is steady motion. It means the object is not moving at all.
• Thinking that a velocity-time graph is a picture of the path. A line going up means speeding up. It does not mean the object is climbing a hill.

Think it through

Write your answers in full sentences. Try each one before you read the pointers.

1. A distance-time graph is a flat horizontal line. A friend says, "The object is moving at a steady speed." Explain what is wrong, and say what the line really shows.
2. The velocity-time graphs of two cars cross at 10 s. Does this mean that the cars meet at 10 s? Say what the crossing point does tell you, and what else you would need to know.
3. A lift starts from rest on the ground floor, goes up and stops at the fifth floor. Describe the shape of its velocity-time graph, and explain each part in terms of what the lift is doing.
4. Explain why the area under a velocity-time graph gives the distance. Use the idea of distance = velocity × time for a very short time.
5. You have a stopwatch and marks on a road every 10 m. Describe how you would collect data to draw the distance-time graph of a cyclist. How would the graph show that the cyclist changed speed?

Pointers for your answers

1. The distance is not changing, so the object is at rest. Speed is the slope, and the slope here is zero.
2. Crossing means that the two cars have the same velocity at 10 s. It says nothing about where they are. You would need their starting positions, or the areas under both graphs.
3. The line rises (speeding up), may stay flat for a while (steady velocity), and then falls to zero (slowing down). The area under the line is the height of the five floors.
4. In a very short time the velocity hardly changes, so velocity × time is a thin rectangle. The whole area is the sum of many thin rectangles.
5. Note the time at each mark. Plot distance against time. A straight line means a steady speed. A change in the steepness shows a change of speed.