# 3.1.3 Displacement & Velocity-Time Graphs

### Displacement-Time Graphs

• Displacement-time graphs show the changing position of an object in motion
• They also show whether an object is moving forwards (positive displacement) or backwards (negative displacement)

Velocity = Gradient of a displacement-time graph

• The greater the slope, the greater the velocity
• A negative gradient = a negative velocity (the object is moving backwards)

#### Worked Example

A car driver sees a hazard ahead and applies the brakes to bring the car to rest.

What does the displacement-time graph look like?  ### Velocity-Time Graphs

• Velocity-time graphs show the speed and direction of an object in motion over a specific period of time
• The area under a velocity-time graph is equal to the displacement of a moving object

Displacement = Area under a velocity-time graph

• Acceleration is any change in the velocity of an object in a given time
• As velocity is a vector quantity, this means that if the speed of an object changes, or its direction changes, then it is accelerating
• An object that slows down tends to be described as ‘decelerating’

Acceleration = Gradient of a velocity-time graph

#### 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. #### Summary of Gradients & Areas

• The gradient of a displacement-time graph is the velocity
• The gradient of a velocity-time graph is the acceleration
• The area under a velocity-time graph is the displacement
• The area under an acceleration-time graph is the velocity #### 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!

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