# 4.3.2 Motion Graphs

### Motion Graphs

• The gradient of a displacement-time graph is the velocity
• The gradient of a velocity-time graph is the acceleration

#### Area Under the Graph

• The area under a velocity-time graph is the displacement
• The area under an acceleration-time graph is the velocity

#### Motion of a Bouncing Ball

• For a bouncing ball, the acceleration due to gravity is always in the same direction (in a uniform gravitational field such as the Earth’s surface)
• This is assuming there are no other forces on the ball, such as air resistance
• Since the ball changes its direction when it reaches its highest and lowest point, the direction of the velocity will change at these points
• The vector nature of velocity means the ball will sometimes have a:
• Positive velocity if it is travelling in the positive direction
• Negative velocity if it is travelling in the negative direction
• An example could be a ball bouncing from the ground back upwards and back down again
• The positive direction is taken as upwards
• This will be either stated in the question or can be chosen, as long as the direction is consistent throughout
• Ignoring the effect of air resistance, the ball will reach the same height every time before bouncing from the ground again
• When the ball is travelling upwards, it has a positive velocity which slowly decreases (decelerates) until it reaches its highest point

• At point A (the highest point):
• The ball is at its maximum displacement
• The ball momentarily has zero velocity
• The velocity changes from positive to negative as the ball changes direction
• The acceleration, g, is still constant and directed vertically downwards
• At point B (the lowest point):
• The ball is at its minimum displacement (on the ground)
• Its velocity changes instantaneously from negative to positive, but its speed (magnitude) remains the same
• The change in direction causes a momentary acceleration (since acceleration = change in velocity / time)

#### Worked Example

The velocity-time graph of a vehicle travelling with uniform acceleration is shown in the diagram below.

Calculate the displacement of the vehicle at 40 s.

#### Exam Tip

Always check the values given on the y-axis of a motion graph – students often confuse displacement-time graphs and velocity-time graphs.

The area under the graph can often be broken down into triangles, squares and rectangles, so make sure you are comfortable with calculating area!

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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