1.2 Motion – Interpreting Distance‑Time and Speed‑Time Graphs
Learning Objectives
- Define speed, velocity and acceleration and write the required formulae.
- Read and interpret both distance‑time and speed‑time graphs.
- From a data table or from the shape of a speed‑time graph decide whether the acceleration is constant or changing.
- Calculate acceleration from the gradient of a speed‑time graph (including correct units).
- Calculate speed from the gradient of a distance‑time graph and distance from the area under a speed‑time graph.
- Describe free fall with and without air resistance and state the value of the acceleration due to gravity g.
- Use the graphs to predict future speed and to obtain the total distance travelled.
Key Quantities, Definitions & Equations
| Quantity | Definition / Formula | Symbol & Units |
|---|
| Speed (scalar) | Distance travelled per unit time.
Instantaneous: \(v = \dfrac{ds}{dt}\)
Average: \( \displaystyle \bar v = \frac{\Delta s}{\Delta t}\) | v, m s⁻¹ |
| Velocity (vector) | Speed with a direction. Sign (+/–) shows direction along the chosen axis.
Instantaneous: \( \vec v = \dfrac{d\vec s}{dt}\) | \(\vec v\), m s⁻¹ |
| Acceleration (vector) | Rate of change of velocity.
Instantaneous: \( \vec a = \dfrac{d\vec v}{dt}\)
Average: \( \displaystyle \bar a = \frac{\Delta v}{\Delta t}\) | \(\vec a\), m s⁻² |
| Constant‑acceleration equations (linear motion) | - \(v = u + at\)
- \(s = ut + \tfrac12 at^2\)
- \(v^2 = u^2 + 2as\)
(where \(u\) = initial speed, \(v\) = final speed, \(s\) = displacement, \(t\) = time, \(a\) = constant acceleration) | – |
| Acceleration due to gravity | \(g \approx 9.8\ \text{m s}^{-2}\) (downward) | g, m s⁻² |
Reading a Distance‑Time Graph
- Slope (gradient) = speed. A steeper slope means a higher speed.
- Horizontal line (zero slope) = the object is at rest.
- Straight line with constant non‑zero slope = constant speed (zero acceleration).
- Straight line through the origin = constant speed from rest (still zero acceleration).
- Curved line = speed is changing; the curvature tells you whether the acceleration is increasing or decreasing.
- To obtain the instantaneous speed at any point, draw a tangent and read its gradient.
Reading a Speed‑Time Graph
- Gradient = acceleration. A straight‑line graph has a constant gradient → constant acceleration.
- Horizontal line = zero acceleration (speed is constant).
- Area under the graph = distance travelled during that time interval.
- For a graph starting from the origin, a straight line gives a triangular area: \(\displaystyle s = \tfrac12 vt\) (since \(v = at\)).
- If the line is curved, the gradient is changing → acceleration is changing. The distance is found by calculating the area (e.g., using geometry or numerical methods).
Constant vs. Changing Acceleration
| Feature | Constant Acceleration | Changing Acceleration |
|---|
| Speed‑time graph | Straight line (gradient = constant). If the motion starts from rest the line passes through the origin. | Curved line (gradient varies). No single straight‑line fit. |
| Distance‑time graph | Parabolic curve (since \(s = ut + \tfrac12 at^2\)). | More complex curvature; not a simple parabola. |
| Numerical test from data | Successive values of \(\displaystyle \frac{\Delta v}{\Delta t}\) are equal (within experimental error). | Successive \(\displaystyle \frac{\Delta v}{\Delta t}\) differ. |
How to decide from a data table
- Calculate the change in speed \(\Delta v\) for each successive time interval.
- Divide each \(\Delta v\) by the corresponding \(\Delta t\) to obtain the average acceleration for that interval.
- If all the accelerations are the same (to within experimental uncertainty) → constant acceleration.
- If they vary → changing acceleration.
Example – Data Table
| t (s) | v (m s⁻¹) | Δv (m s⁻¹) | Δt (s) | Δv/Δt (m s⁻²) |
|---|
| 0 | 0 | – | – | – |
| 2 | 4 | 4 | 2 | 2.0 |
| 4 | 8 | 4 | 2 | 2.0 |
| 6 | 12 | 4 | 2 | 2.0 |
All calculated accelerations are 2 m s⁻² → the object has constant acceleration.
Example – Curved Speed‑Time Graph (Changing Acceleration)
Suppose the speed increases rapidly at first and then more slowly. A typical data set might be:
| t (s) | v (m s⁻¹) | Δv/Δt (m s⁻²) |
|---|
| 0 | 0 | – |
| 1 | 5 | 5.0 |
| 2 | 9 | 4.0 |
| 3 | 12 | 3.0 |
The decreasing values of \(\Delta v/\Delta t\) show that the acceleration is decreasing – the speed‑time graph is curved.
Free Fall and Terminal Velocity
- In vacuum, a falling object experiences a constant downward acceleration \(g = 9.8\ \text{m s}^{-2}\).
- With air resistance, the upward drag force grows with speed. When the drag equals the weight, the net force is zero and the object stops accelerating – this is terminal velocity (\(v_t\)).
- For an object falling from rest:
- Speed after time \(t\): \(v = gt\) (until \(v\) approaches \(v_t\)).
- Distance fallen: \(s = \tfrac12 gt^2\) (again, only while \(v < v_t\)).
- On a speed‑time graph, free fall in vacuum is a straight line through the origin with gradient \(g\). With air resistance the line starts straight but then curves, approaching a horizontal asymptote at \(v_t\).
Worked Example – Uniform Acceleration
An object starts from rest and accelerates uniformly at \(a = 2.0\ \text{m s}^{-2}\) for \(5.0\ \text{s}\).
- Final speed: \(v = u + at = 0 + (2.0)(5.0) = 10.0\ \text{m s}^{-1}\).
- Distance travelled: \(s = ut + \tfrac12 at^2 = 0 + \tfrac12 (2.0)(5.0)^2 = 25.0\ \text{m}\).
- Speed‑time graph: straight line from (0,0) to (5 s, 10 m s⁻¹). Gradient = 2.0 m s⁻² (constant acceleration).
- Area under the graph (triangle) = \(\tfrac12 \times 5.0\ \text{s} \times 10.0\ \text{m s}^{-1} = 25.0\ \text{m}\) – matches the calculation above.
Worked Example – Changing Acceleration
A car’s speed (m s⁻¹) is recorded every second as follows: 0, 4, 7, 9, 10. Determine the nature of the acceleration.
- Calculate \(\Delta v/\Delta t\) for each 1‑s interval:
- 1 s: \(4/1 = 4\ \text{m s}^{-2}\)
- 2 s: \((7-4)/1 = 3\ \text{m s}^{-2}\)
- 3 s: \((9-7)/1 = 2\ \text{m s}^{-2}\)
- 4 s: \((10-9)/1 = 1\ \text{m s}^{-2}\)
- The acceleration decreases each second, so the acceleration is changing. The speed‑time graph would be a curve that becomes less steep with time.
Summary Checklist
- Speed = distance ÷ time; velocity = speed with direction.
- Acceleration = change in velocity ÷ time.
- Gradient of a distance‑time graph → instantaneous speed.
- Gradient of a speed‑time graph → acceleration.
- Area under a speed‑time graph → distance travelled.
- Constant acceleration → straight line on a speed‑time graph (constant gradient) and a parabolic distance‑time graph.
- Changing acceleration → curved speed‑time graph; gradient varies; distance‑time graph is not a simple parabola.
- Free fall in vacuum: \(a = g = 9.8\ \text{m s}^{-2}\). With air resistance the speed‑time graph approaches a horizontal asymptote (terminal velocity).