1.2 Motion – Determining Acceleration from Speed‑Time Graphs
Learning Objective
From given data or the shape of a speed‑time graph, decide whether an object is moving with:
Constant acceleration
Changing acceleration
Key Concepts
Acceleration \$a\$ is the rate of change of speed with time: \$a=\frac{\Delta v}{\Delta t}\$
On a speed‑time graph:
A straight‑line segment indicates a constant rate of change of speed (constant acceleration).
A curved segment indicates that the rate of change of speed is itself changing (changing acceleration).
The gradient (slope) of the graph at any point gives the instantaneous acceleration.
Identifying Constant Acceleration
Characteristics of a speed‑time graph for constant acceleration:
The graph is a straight line (inclined or horizontal).
The gradient is the same over the whole interval.
If the line passes through the origin, the object started from rest.
Mathematical relationship (derived from the definition of acceleration):
\$v = u + at\$
where \$u\$ is the initial speed, \$v\$ the speed after time \$t\$, and \$a\$ the constant acceleration.
Identifying Changing Acceleration
Characteristics of a speed‑time graph for changing acceleration:
The graph is curved (concave up or down).
The gradient varies with time; therefore \$a\$ is not constant.
Common causes: varying net force, drag, or non‑uniform propulsion.
When acceleration changes, the speed‑time relationship cannot be expressed by a single linear equation. Instead, the instantaneous acceleration at any time \$t\$ is given by the derivative:
\$a(t)=\frac{dv}{dt}\$
Worked Example – Interpreting Data
Consider the following experimental data for a cart moving in a straight line:
Time \$t\$ (s)
Speed \$v\$ (m s⁻¹)
0
0
1
2
2
4
3
6
4
8
Calculate the acceleration between each successive pair of points and decide whether it is constant.
Between \$t=0\$ s and \$t=1\$ s: \$a=\frac{2-0}{1-0}=2\ \text{m s}^{-2}\$
Between \$t=1\$ s and \$t=2\$ s: \$a=\frac{4-2}{2-1}=2\ \text{m s}^{-2}\$
Between \$t=2\$ s and \$t=3\$ s: \$a=\frac{6-4}{3-2}=2\ \text{m s}^{-2}\$
Between \$t=3\$ s and \$t=4\$ s: \$a=\frac{8-6}{4-3}=2\ \text{m s}^{-2}\$
All calculated accelerations are equal; therefore the motion has constant acceleration. The corresponding speed‑time graph would be a straight line through the origin with a gradient of \$2\ \text{m s}^{-2}\$.
Worked Example – Curved Graph
Data for a falling object (including air resistance) are:
Time \$t\$ (s)
Speed \$v\$ (m s⁻¹)
0
0
1
9
2
16
3
21
4
24
Accelerations:
\$a_{0-1}=9\ \text{m s}^{-2}\$
\$a_{1-2}=7\ \text{m s}^{-2}\$
\$a_{2-3}=5\ \text{m s}^{-2}\$
\$a_{3-4}=3\ \text{m s}^{-2}\$
The acceleration decreases with time, indicating changing acceleration. The speed‑time graph would be a curve that becomes less steep as time increases.
How to Analyse a Speed‑Time Graph Quickly
Look at the overall shape:
Straight line → constant acceleration.
Curved line → changing acceleration.
Check the gradient:
Same gradient throughout → constant \$a\$.
Gradient varies → \$a\$ is changing.
If the line is horizontal (gradient \$=0\$), the acceleration is zero and the object moves at constant speed.
If the line passes through the origin with a constant gradient, the object started from rest with constant acceleration.
Common Pitfalls
Confusing a straight‑line speed‑time graph with a constant speed graph. A horizontal line (zero gradient) means constant speed; a sloping straight line means constant acceleration.
Assuming a curved graph always indicates decreasing speed. The curvature only tells you that the acceleration is changing; the speed may still be increasing.
Neglecting units when calculating gradients; always express acceleration in \$\text{m s}^{-2}\$.
Suggested Practice Questions
Sketch a speed‑time graph for an object that starts from rest, accelerates uniformly for 3 s, then moves at constant speed for another 2 s.
Given the graph below (describe verbally: a parabola opening upwards starting at the origin), state whether the acceleration is constant, increasing, or decreasing.
Calculate the acceleration at \$t=5\,\$s for a speed‑time graph described by \$v=4t^2\$.
Suggested diagram: Sketch of a straight‑line speed‑time graph (constant acceleration) and a curved speed‑time graph (changing acceleration) placed side by side for comparison.
Summary
To determine the nature of acceleration from a speed‑time graph:
Identify whether the graph is a straight line (constant \$a\$) or curved (changing \$a\$).
Use the gradient to find the numerical value of acceleration.
Remember that a horizontal line means zero acceleration (constant speed).