Travel graphs: Distance-time graphs

​​In a nutshell

Travel graphs show distance travelled on the $$y$$-axis against time on the $$x$$-axis. They can be used to determine the speed and direction of motion.

What do travel graphs look like?

Travel graphs have a series of line segments that join together to represent a journey. An example travel graph is given below. It shows a journey to a supermarket and back, with a small stop on the way to the shop.

The graph can be broken into its separate straight line segments:

• ($$0$$​ mins to $$10$$ mins) In the first ten minutes, the walker has travelled one kilometre;
  
• ($$10$$​ mins to $$15$$​ mins) then the person stops for five minutes. Perhaps they are chatting to someone on the way to the shop;
• ($$15$$​ mins to $$20$$​ mins) the person continues on their journey to the shop, travelling $$0.6$$ kilometres in five minutes;
• ($$20$$​ mins to $$35$$​ mins) the person has stopped at the shop to do their shopping. They don't go any further for $$15$$ minutes. With respect to their journey from home, they have stopped;
• ($$35$$ mins to $$50$$ mins) the person has finished their shopping and goes directly home. They travel $$1.6$$ kilometres in $$15$$ minutes.

​​Direction

If the graph is going up, then the direction is away from the starting position. If the graph is going down, then the direction is toward the starting position. If the graph is flat, then motion has stopped.

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Speed

Travel graphs give you enough information to calculate the speed of the motion. 

The speed of a segment is calculated by finding the gradient of the line segment. Gradient is calculated with the formula:

​$$m=\frac{\text{change in }y}{\text{change in }x}$$

Note: If a line segment has a negative gradient, ignore the negative sign for the speed. The negative just means that motion is back towards the starting position.

Given that each line segment is either a diagonal line or a flat line, finding the gradient is straightforward: for diagonal lines, just use the coordinates of the top and bottom of the line; for flat lines, the gradient, and hence speed, is zero. 

Note: The straight line means that the speed is constant.

Alternatively, use:

​$$\text{speed}=\frac{\text{distance}}{\text{time}}$$​​

Note: The unit of speed will the unit of distance divided by the unit of time.

Example 1

In the first ten minutes of the walker's journey, how fast do they travel?

In the first ten minutes, the walker has travelled one kilometre. Hence their speed is:

$$\frac{1\text{ km}}{10\text{ mins}}=0.1\text{ km per minute} = \underline{100\text{ m per minute}}$$

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Example 2

How fast did they walk on their way back home? Use the same graph above again.

Use the final $$15$$ minutes of the journey; the graph is going down, hence the direction is back toward the starting location. In this time, the walker travels $$1.6\text{ km}.$$ Hence their speed (rounded to three significant figures) was:

$$\frac{1.6\text{ km}}{15\text{ mins}}=0.107\text{ km per minute} = \underline{107\text{ m per minute}}$$​​