3.1.3 Displacement & Velocity-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)

• 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)

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