11.2 - Linear Motion Graphs

Linear motion is represented using displacement-time graphs and velocity-time graphs.

Displacement-Time Graph

• Vertical axis: displacement, $s$ (m); horizontal axis: time, $t$ (s).
• The gradient gives velocity.

Graph feature	Interpretation
Straight line with positive gradient	Uniform velocity in the positive direction
Steeper positive gradient	Greater uniform velocity
Horizontal line	Object is stationary; velocity is zero
Straight line with negative gradient	Uniform velocity in the opposite direction

• Between $(t_1,s_1)$ and $(t_2,s_2)$: $v=\frac{s_2-s_1}{t_2-t_1}$.

Velocity-Time Graph

• Vertical axis: velocity, $v$ (${\text{m s}}^{-1}$); horizontal axis: time, $t$ (s).
• The gradient gives acceleration: $a=\frac{v-u}{t}$.
• The area under the graph gives displacement.

Graph feature	Interpretation
Straight line with positive gradient	Velocity increases uniformly; uniform acceleration
Horizontal line above the time axis	Constant velocity; zero acceleration
Straight line with negative gradient	Velocity decreases uniformly; uniform deceleration
Line on the time axis	Zero velocity; object is stationary

For the motorcycle graph:

• OA: positive uniform gradient → velocity increases uniformly; uniform acceleration.
• AB: zero gradient → constant velocity; zero acceleration.
• BC: negative uniform gradient → velocity decreases uniformly; uniform deceleration.
• CD: line on the time axis → motorcycle is stationary; zero acceleration.

Numerical Problems from Motion Graphs

Displacement-time graph: read displacement from the vertical axis and calculate velocity from the gradient of the required section.

Velocity-time graph: read velocity from the vertical axis, calculate acceleration from the gradient and calculate displacement by adding the areas under the graph.

• Average velocity: $\text{average velocity}=\frac{\text{total displacement}}{\text{total time}}$.
• Rectangle: $A=\text{base}\times \text{height}$
• Triangle: $A=\frac{1}{2}\times \text{base}\times \text{height}$
• Trapezium: $A=\frac{1}{2}\times \text{sum of parallel sides}\times \text{height}$
• Correct S.I. units: displacement in m, velocity in ${\text{m s}}^{-1}$ and acceleration in ${\text{m s}}^{-2}$.

Activity 11.4: Drawing and Interpreting Graphs

Aim: Draw and interpret displacement-time and velocity-time graphs.

1. Work in groups and select a coloured envelope prepared by the teacher.
2. Discuss the situation in the selected envelope and draw its graph.
3. Present the result to the class.