Section 1.3: Displacement-Time & Velocity-Time Graphs

Graphs are powerful tools for representing motion. Two of the most common are the displacement-time graph and the velocity-time graph. These allow us to visualize how objects move over time.

Displacement-Time Graphs

The slope of a displacement-time (\(x\)-\(t\)) graph gives the velocity of the object:

  • Straight line with positive slope → constant positive velocity.
  • Straight line with negative slope → constant negative velocity.
  • Curved line → changing velocity (acceleration).

Key: Slope of displacement-time graph = velocity.

Velocity-Time Graphs

The slope of a velocity-time (\(v\)-\(t\)) graph gives the acceleration, while the area under the graph gives the displacement.

  • Horizontal line → constant velocity (zero acceleration).
  • Positive slope → positive acceleration.
  • Negative slope → negative acceleration (deceleration).

Key: Slope = acceleration, Area under curve = displacement.

Example 1

A car travels with constant velocity of 20 m/s for 5 seconds. Sketch the displacement-time and velocity-time graphs.

Displacement-Time Graph: Straight line with slope \(20\). At \(t=5\) s, displacement \(= 100\) m.

Velocity-Time Graph: Horizontal line at \(v=20\) m/s from \(t=0\) to \(t=5\).

Example 2

A particle starts from rest and accelerates uniformly at \(2 \,\text{m/s}^2\) for 4 seconds. Sketch the velocity-time graph and calculate displacement.

Velocity after 4 s: \(v = at = 2 \times 4 = 8 \,\text{m/s}\).

Velocity-time graph: Straight line from (0,0) to (4,8).

Displacement = area under graph = \(\tfrac{1}{2} \times 4 \times 8 = 16 \,\text{m}\).

Practice Problems

  1. A body moves with velocity 5 m/s for 6 s. Sketch displacement-time and velocity-time graphs. Find displacement.
  2. A car accelerates uniformly from 10 m/s to 30 m/s in 5 s. Draw the velocity-time graph and calculate displacement.
  3. An object starts from rest and accelerates at 3 m/s² for 2 s. Find its velocity and displacement.
  4. A ball moves at constant velocity of -4 m/s for 3 s. Sketch both graphs.
  5. A runner’s velocity-time graph is a triangle from (0,0) to (10,8) and back to (20,0). Find total displacement.
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