Gradients of real life graphs

​​In a nutshell

Gradients of real life graphs represent the rate of change of the $$y$$-variable with respect to the $$x$$-variable.

Gradient reminder

Recall that

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

Note on units compatibility

When using the gradient of a real life graph, it's important that the units on the axes are compatible. This means that if both axes are using some form of measurement unit, they use the same unit for that measurement. For example, in a velocity-time graph, velocity has a unit of distance unit divided by time unit. This time unit must be the same unit used on the $$x$$-axis. 

Example 1

A velocity-time graph has on the $$y$$-axis (velocity axis) a unit of metres per second. What unit must the $$x$$-axis (time axis) have?

Since the velocity unit is metres per second, it follows that the time unit must be seconds.

If the axes' units are not compatible, the unit of the gradient won't make sense.

Example 2

What unit is obtained by dividing a velocity with unit km/h by a time with unit hours?

You can calculate this unit as if algebraically:

$$\text{unit}=\frac{\frac{\text{km}}{\text{h}}}{\text{h}}=\frac{\text{km}}{\text{h}^2}$$

This is kilometres per hour squared, which is a standard unit of acceleration.

Example 3

What unit is obtained by dividing a velocity with unit km/h by a time with unit seconds?

Again, you can calculate this unit as if algebraically:

$$\text{unit}=\frac{\frac{\text{km}}{\text{h}}}{\text{s}}=\frac{\text{km}}{\text{h}\times\text{s}}$$

This is kilometres per hours seconds, which is not a useful, nor used, unit. This happened because the units were not compatible.

The gradient on a distance-time graph

The gradient of a distance-time graph represents the velocity of that segment's motion, if the axes' units are compatible.

Distance is on the $$y$$-axis and time is on the $$x$$-axis, so it follows from

$$\text{speed}=\frac{\text{change in distance}}{\text{change in time}}$$

that gradient gives velocity. 

Note: A negative gradient represents that the motion is in the direction back towards the beginning. 

The gradient on a velocity-time graph

As long as your axes' units are compatible (using the same time unit), the gradient of a velocity-time graph represents the acceleration of that segment's motion. Acceleration is the rate of change of velocity, so is a measure of how velocity is changing.

​

Distance is on the $$y$$-axis and time is on the $$x$$-axis, so it follows from

$$\text{acceleration}=\frac{\text{change in velocity}}{\text{change in time}}$$

that gradient gives acceleration. 

Note: A negative gradient shows that the motion is slowing down, not that motion is in the direction back towards the beginning. Velocity can still be positive even if the gradient at that point is negative.

A flat line on a velocity-time graph represents no change in velocity. In other words, motion occurs at a constant speed. Don't get this confused with a flat line on distance-time graph which means motion has stopped.

The gradient of a general real life graph

As seen, the gradient of a graph has a unit calculated by the $$y$$-unit divided by the $$x$$-unit. Sometimes this gives an unhelpful unit, but other times it gives a recognised unit.

Example 4

A force-area graph has the force unit Newtons on the $$y$$-axis and the area unit metres squared on the $$x$$-axis. What does the gradient represent?

The gradient's unit is given by the $$y$$ unit divided by the $$x$$-unit:

$$\text{unit}=\frac{\text{N}}{\text{m}^2}=\text{Nm}^{-2}$$​​

This unit, Newtons per metres squared is better known as pascals, $$\text{Pa}$$, and is a unit of pressure.