Displacement-Time Graphs
The motion of an object can be translated into a graph from which we can find more information about the journey. The graph below shows a displacement-time graph:
We can break down what is happening at each point:
1 – Constant Velocity, Positive Direction – a straight line indicates a constant velocity. Here it is in the positive direction, for example a runner going forwards away from their house. (remember displacement and velocity are vector quantities and have direction).
2 – Stationary – a horizontal line indicates an object is stationary, for example a runner taking a break.
3 – Constant Velocity, Negative Direction – again this straight line indicates constant velocity, however here it is in the negative direction, for example a runner could be running back towards their house.
4 – Acceleration – a curved line indicates an object is accelerating, for example a runner speeding up to catch up with a friend.
Gradient of a Displacement-Time Graph
From the graph we can calculate the velocity of the object at different points.
Velocity is equal to the displacement of an object per unit time. We also know that the gradient of a graph can be calculated from the change in y divided by the change in x on a graph.
In a displacement-time graph we have displacement on the y-axis and time on the x-axis.
Therefore if we calculate the gradient we are actually calculating the velocity.
When the line on a displacement-time graph is straight (like points 1 and 3) calculating the gradient gives us the constant velocity of the object for that time period. The steeper the line (gradient) the faster the velocity of the object.
A horizontal line (like point 2) gives a gradient, and therefore velocity, of 0.
We can calculate the gradient at any point on a curved line (like point 4) to give us the instantaneous velocity at that point.
It is important to note that if a curved line begins to get less steep, the object is decelerating.
Velocity-Time Graphs
The displacement-time graph above can be translated into a velocity-time graph to give the following:
This graph shows exactly the same information about the journey of the object:
1 – Constant Velocity, Positive Direction
2 – Stationary
3 – Constant Velocity, Negative Direction
4 – Acceleration
IMPORTANT: The motion above the x-axis represents the object moving in the positive direction, i.e. forwards, and the motion below the x-axis represents the object moving in the negative direction, i.e. backwards.
Gradient of a Velocity-Time Graph
Like the displacement-time graph the gradient of a velocity-time graph can also give us some useful information.
Acceleration is equal to the rate of change in velocity. From the velocity-time graph we can see that we have velocity on the y-axis and time on the x-axis.
Therefore, if we calculate the gradient of a velocity-time graph we get the acceleration.
If the velocity is constant we know that the acceleration must be zero and therefore (like at points 1, 2 and 3 the in the graph) the gradient must be 0.
Graph 3 highlights the differences between acceleration and deceleration:
1 – Acceleration in the positive direction (positive gradient, above the x-axis).
2 – Acceleration in the negative direction (positive gradient, below the x-axis).
3 – Deceleration in the positive direction (negative gradient, above the x-axis).
4 – Deceleration in the negative direction (negative gradient, below the x-axis).
Area of a Velocity-Time Graph
The gradient isn’t the only useful piece of information we can get from a velocity-time graph.
The area under a velocity-time graph gives us the displacement of the object.
This is true because displacement is equal to the product of velocity and time. As velocity is on the y-axis and time is on the x-axis, calculating the are under the cure is the same as multiplying these quantities together and so we get the displacement.
The easiest way to calculate the total displacement of an object is to split the area under the graph up into simple shapes. For example in graph 2 we can split it up into a rectangle, a square and a triangle:
Adding up the area of the shapes will give us the total displacement of the object.
Key Points
- The gradient of a displacement-time graph gives the velocity.
- The gradient of a velocity-time graph gives the acceleration.
- The area under a velocity-time graph gives the displacement.
Worked Examples
Example 1
A runner goes for a run staring from their house. Her journey is represented by the displacement-time graph below. Describe her run.
Example 2
The velocity-time graph below shows the the journey of a car. Calculate the acceleration of the car between points A-B and C-D and state whether the car is travelling forwards or backwards.
Please leave any questions in the comments below and for a simple explanation on the basics of kinematics click here. To keep up to date on the latest posts follow me on Pinterest or Instagram.