# 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.

##

## 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:*

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

**

##### 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:*

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