Velocity-time graphs

Velocity-time graphs will give us a value for velocity in relation to time. These can easily be used to find things like the acceleration of an object and the distance travelled.

Velocity-time graphs are very similar to the distance-time graphs previously covered. In a velocity-time graph time is plotted on the  axis but instead of distance, velocity is plotted on the  axis.

Like a displacement-time graph, velocity-time graphs can have a negative gradient if there is a decrease in the velocity.

The graph above shows that we have an increasing velocity for the first 2 seconds and then this stays the same for 3 seconds. Then the line starts to go down with a negative gradient. This does not mean that we are moving backwards as we are still travelling in the same direction, just at a decreasing velocity. This graph would be the equivalent of getting in a car and speeding up to a constant speed of 40 metres per second for 3 seconds and then applying the brakes so that we slow down before coming to an eventual stop after 7 seconds.

Gradients on velocity-time graphs

The gradient on a velocity-time graph can be used to determine the acceleration of the object. When the line on the graph is flat this shows 0 acceleration (as we are at a constant speed). A positive gradient shows acceleration, as the velocity is increasing, and a negative gradient shows deceleration, as the velocity is decreasing.

To determine the numerical value for acceleration using the gradient of the graph, you divide the  axis value by the  axis value.

Example

Find the following from the graph below:

The acceleration at 2 seconds.
The acceleration at 6 seconds.

1.  At 2 seconds the gradient of the line is such that for every second that passes, the velocity increases by 5 metres per second. Therefore, the acceleration must be 5 ms^(-2).
2.  At 6 seconds we are decelerating at a rate of 13.33 ms^(-2) therefore the acceleration is -13.33 ms^(-2)

Determining the distance travelled

Velocity-time graphs can also be used to determine the distance travelled. To do this we simply work out the area under the line of the graph. When doing this we should have a simple shape that can then be examined and an area found which should be left in the units that are found in the question. This is shown for the example on the previous page:

The area under the graph can be split up into a rectangle and two triangles to have an area of 200. Therefore, this graph shows a distance moved of 200 metres.

Finding the distance from a velocity-time graph really is as easy as finding the area (from the numbers presented on the axes) that is under the line. This can obviously then be split up into sections so we could find the distance travelled up to 4 seconds, for example, or the distance travelled between the 3^rd and 6^th second. All that we have to do is to find the area under the line between these two points on the  axis.

Example

Find the following from the graph below:

1. The total distance travelled.
2. The distance travelled between 3 and 6 seconds.

1. To find the total distance travelled we need to work out the area underneath the line.
   The above graph is easily broken up into sections that are either rectangular or triangles so that each area can be found and then the total added up to give the entire area under the graph. Doing this we calculate a total distance of 110 m.

2. To calculate the distance travelled between 3 and 6 seconds, we work out the area under the graph but only between 3 and 6 seconds. This is fairly easy to do, and again the area is split down into manageable sections and then added together.

This gives us a distance travelled between 3 and 6 seconds of 80.5 m.

Summary of velocity-time graphs