Sketch, plot and interpret distance-time and speed-time graphs

Cambridge IGCSE Physics 0625 – Motion

Objective

Students will be able to sketch, plot and interpret distance‑time and speed‑time graphs.

Key Concepts

• Distance (\$s\$) – total length of the path travelled.
• Displacement – straight‑line distance from start to finish (not required for this objective).
• Speed (\$v\$) – rate of change of distance: \$v = \frac{s}{t}\$
• Velocity – speed with a direction (used when interpreting speed‑time graphs).
• Acceleration (\$a\$) – rate of change of speed: \$a = \frac{\Delta v}{\Delta t}\$
• Gradient of a graph – represents a physical quantity (e.g., slope of a distance‑time graph = speed).

Sketching Distance‑Time Graphs

A distance‑time graph shows how far an object has travelled from a fixed point over time.

• Horizontal axis (x‑axis): time (\$t\$) in seconds (s).
• Vertical axis (y‑axis): distance (\$s\$) in metres (m).
• A straight, horizontal line indicates the object is at rest.
• A straight line with a constant positive gradient indicates uniform speed.
• A curved line indicates changing speed (acceleration or deceleration).

Sketching Speed‑Time Graphs

A speed‑time graph shows how the speed of an object varies with time.

• Horizontal axis: time (\$t\$) in seconds (s).
• Vertical axis: speed (\$v\$) in metres per second (m s⁻¹).
• A horizontal line through the origin represents an object at rest.
• A horizontal line above the time axis indicates constant speed.
• A straight line with a positive gradient indicates constant acceleration.
• The area under the curve gives the distance travelled: \$\text{distance} = \int v\,dt\$ (for straight‑line sections, distance = speed × time).

Plotting Real Data

When plotting experimental data, follow these steps:

1. Label the axes clearly with quantity and units.
2. Choose an appropriate scale so the data fill the graph area.
3. Mark each data point accurately.
4. Connect points with a smooth line (do not use a ruler for curved sections).
5. Draw a best‑fit straight line where the motion is uniform.

Interpreting Graphs

Use the following table to link graph features with physical meanings.

Sample Calculations

1. An object moves with a constant speed of 4 m s⁻¹ for 5 s. What distance does it travel?

\$\text{distance}=v\times t = 4\;\text{m s}^{-1}\times5\;\text{s}=20\;\text{m}\$

2. A speed‑time graph shows a straight line from (0 s, 0 m s⁻¹) to (3 s, 9 m s⁻¹). Determine the acceleration.

\$a=\frac{\Delta v}{\Delta t}= \frac{9-0}{3-0}=3\;\text{m s}^{-2}\$

3. Using the same graph, find the distance travelled during the 3 s.

The area under the line is a triangle:

\$\text{distance}= \frac{1}{2}\times\text{base}\times\text{height}= \frac{1}{2}\times3\;\text{s}\times9\;\text{m s}^{-1}=13.5\;\text{m}\$

Common Misconceptions

• Confusing the slope of a speed‑time graph (acceleration) with the area under it (distance).
• Assuming a curved distance‑time graph always means deceleration; it could be acceleration or a change in speed direction.
• Reading the vertical axis of a distance‑time graph as speed – the gradient, not the axis value, gives speed.

Practice Questions

1. Sketch a distance‑time graph for a car that starts from rest, accelerates uniformly for 4 s to a speed of 12 m s⁻¹, then travels at that speed for a further 6 s.
2. From the graph you have drawn, determine:

   1. The acceleration during the first 4 s.
   2. The total distance travelled after 10 s.

3. A speed‑time graph shows a constant speed of 5 m s⁻¹ for 8 s, then a uniform deceleration to rest in 2 s. Calculate the distance covered during the deceleration phase.

Summary

Understanding distance‑time and speed‑time graphs is fundamental for analysing motion. The gradient of a distance‑time graph gives speed, while the gradient of a speed‑time graph gives acceleration. The area under a speed‑time graph represents the distance travelled. Accurate sketching, plotting, and interpretation of these graphs enable students to solve a wide range of physics problems.