2.1.4 Graphs Describing Motion

• Three types of graphs that can represent motion are
  • Displacement-time graphs
  • Velocity-time graphs
  • Acceleration-time graphs

Displacement-Time Graph

• On a displacement-time graph…
  • Slope equals velocity
  • The y-intercept equals the initial displacement
  • A straight (diagonal) line represents a constant velocity
  • A curved line represents an acceleration
  • A positive slope represents motion in the positive direction
  • A negative slope represents motion in the negative direction
  • A zero slope (horizontal line) represents a state of rest
  • The area under the curve is meaningless

Displacement-time graphs displacing difference velocities

Velocity-Time Graph

• On a velocity-time graph…
  • Slope equals acceleration
  • The y-intercept equals the initial velocity
  • A straight (diagonal) line represents uniform acceleration
  • A curved line represents non-uniform acceleration
  • A positive slope represents acceleration in the positive direction
  • A negative slope represents acceleration in the negative direction
  • A zero slope (horizontal line) represents motion with constant velocity
  • The area under the curve equals the change in displacement

Velocity-time graphs displacing different acceleration

Acceleration-Time Graph

• On an acceleration-time graph…
  • Slope is meaningless
  • The y-intercept equals the initial acceleration
  • A zero slope (horizontal line) represents an object undergoing constant acceleration
  • The area under the curve equals the change in velocity

How displacement, velocity and acceleration graphs relate to each other

Part (a)

Step 1: Recall that the slope of a velocity-time graph represents the magnitude of acceleration

• • • The slope of a velocity-time graph indicates the magnitude of acceleration

      Therefore, the only sections of the graph where Tora is accelerating is section B and section D

    • Sections A, C, and E are flat – in other words, Tora is moving at a constant velocity (i.e. not accelerating)

Step 2: Identify the section with the steepest slope

• • • Section D of the graph has the steepest slope
      Hence, the largest acceleration is shown in section D

Part (b)

Step 1: Recall that the gradient of a velocity-time graph gives the acceleration

• • • Calculating the gradient of a slope on a velocity-time graph gives the acceleration for that time period

Step 2: Draw a large gradient triangle at the appropriate section of the graph

• • • A gradient triangle is drawn for the time period between 5 and 10 seconds below:

Step 3: Calculate the size of the gradient and state this as the acceleration

• • • The acceleration is given by the gradient, which can be calculated using:

• • • Therefore, Tora accelerated at 1 m s⁻² between 5 and 10 seconds

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 traveling in the positive direction
  • Negative velocity if it is traveling 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 traveling 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)