Question
Draw distance-time graphs for (a) uniform motion and (b) non-uniform motion. Explain what the shape of the graph tells us about the nature of motion.
Solution — Step by Step
A distance-time graph plots distance (on the y-axis) against time (on the x-axis). The slope of this graph at any point gives the speed of the object at that instant.
This is the central idea: the shape of the graph is a visual representation of how speed changes with time.
Uniform motion means the object covers equal distances in equal intervals of time — constant speed.
For uniform motion, (where is constant). This is a linear equation, so the graph is a straight line starting from the origin (if motion begins from rest position at ).
Key features of the graph:
- Straight line
- Constant slope = constant speed
- Steeper line = higher speed (covers more distance in same time)
- Less steep line = lower speed
Example: A car travelling at a constant 60 km/h on a highway. After 1 hour: 60 km. After 2 hours: 120 km. After 3 hours: 180 km. Plot these points → straight line.
Non-uniform motion means the object does not cover equal distances in equal time intervals — speed is changing.
The graph is a curve (not a straight line).
Two important cases:
Accelerating (speeding up): Distance increases at a faster and faster rate. The curve bends upward — getting steeper. The slope (instantaneous speed) increases with time.
Decelerating (slowing down): Distance still increases (object still moving forward) but at a slower and slower rate. The curve flattens out — getting less steep. The slope decreases with time.
A stationary object is a special case: horizontal straight line (distance doesn’t change, slope = 0).
| Shape of d-t graph | Meaning |
|---|---|
| Straight line through origin | Uniform motion from rest position |
| Straight horizontal line | Object is stationary |
| Straight line with steeper slope | Faster uniform motion |
| Upward curving line (concave up) | Accelerating (non-uniform, speeding up) |
| Flattening curve (concave down) | Decelerating (non-uniform, slowing down) |
| Perfectly vertical line | Impossible (infinite speed) |
The slope at any specific point on a curve = instantaneous speed at that moment (measured by drawing a tangent at that point).
Why This Works
The mathematical connection is: (instantaneous speed = derivative of distance with respect to time). The graph is literally a plot of a function, and the slope is its derivative.
For uniform motion, → straight line (linear function).
For non-uniform motion, varies with time → curved graph (non-linear function).
This is why graphical methods are powerful: instead of algebraic equations, the shape tells you the physics directly. Any time you see a curved d-t graph in an exam, immediately say “non-uniform motion” — the speed is changing.
Alternative Method
Numerical approach to verify: Tabulate data for a moving object:
| Time (s) | Distance (m) — Uniform | Distance (m) — Non-uniform |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 5 | 1 |
| 2 | 10 | 4 |
| 3 | 15 | 9 |
| 4 | 20 | 16 |
The uniform case (5 m every second) plots as a straight line. The non-uniform case (distances are 1, 4, 9, 16 — squares of time, i.e., ) plots as an upward curving parabola.
Common Mistake
A very frequent error is drawing a distance-time graph that goes downward after some point. A d-t graph can only slope upward (or be horizontal if stationary) — distance is always increasing or constant, it never decreases. A displacement-time graph CAN slope downward (when an object returns), but distance (total path covered) never decreases. Make sure you’re drawing a distance graph, not a displacement graph, if the question specifies “distance.”
When asked to find speed from a d-t graph, draw a tangent to the curve at the required point and calculate its slope (rise/run). For a straight line graph, the slope is constant everywhere — use any two points.