Motion Graphs (DP IB Physics) : Revision Note

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Motion Graphs

  • The motion of objects can be analysed in terms of position, velocity and acceleration

    • These are all related to each other by gradients and areas under curves

  • Three types of graphs that can represent the motion of an object are:

    • Displacement-time graphs

    • Velocity-time graphs

    • Acceleration-time graphs

Displacement-Time Graphs

  • 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

      Motion graphs (1), downloadable AS & A Level Physics revision notes

Displacement-time graphs displacing different values of velocity

Velocity-Time Graphs

  • 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

      Motion graphs (2), downloadable AS & A Level Physics revision notes

Velocity-time graphs displacing different values of acceleration

Acceleration-Time Graphs

  • 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

Motion graphs (3), downloadable AS & A Level Physics revision notes

How displacement, velocity and acceleration graphs relate to each other

  • Acceleration can either be

    • Uniform i.e. a constant value. For example, acceleration due to gravity on Earth

    • Non-uniform i.e. a changing value. For example, an object with increasing acceleration

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 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

Motion of Bouncing Ball 1, downloadable AS & A Level Physics revision notes
Motion of Bouncing Ball 2, downloadable AS & A Level Physics revision notes
  • 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 accelerationg, 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

Tora is training for a cycling tournament.

The velocity-time graph below shows her motion as she cycles along a flat, straight road.

WE V-T graph Question image, downloadable IGCSE & GCSE Physics revision notes

(a) In which section (A, B, C, D, or E) of the velocity-time graph is Tora’s acceleration the largest? 

(b) Calculate Tora’s acceleration between 5 and 10 seconds.

Answer:

(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

(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:

WE V-T graph Solution image, downloadable IGCSE & GCSE Physics revision notes

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:

a c c e l e r a t i o n space equals space g r a d i e n t space equals space 5 over 5 space equals space 1 space straight m space straight s to the power of negative 2 end exponent

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

Worked Example

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

v-t Area Worked Example (1)

Calculate the displacement of the vehicle at 40 s.

Answer:

v-t Area Worked Example (2), downloadable AS & A Level Physics revision notes
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