Velocity-Time Graphs and Displacement
~8 min read
- Equations: v = u + at, s = ut + ½at², v² = u² + 2as. For uniform acceleration only.
- v-t graph: Slope = acceleration. Area under curve = displacement.
- s-t graph: Slope = velocity. Straight line for uniform motion; parabola for uniform acceleration.
Graphs are the most efficient language of motion. CDS/OTA tests reading v-t and s-t graphs, identifying motion type, and computing displacement from area.
Equations of Motion
For an object with uniform acceleration a, initial velocity u, final velocity v after time t and displacement s:
- v = u + at
- s = ut + ½at²
- v² = u² + 2as
For free fall, a = g = 9.8 m/s² (downward). If thrown up, take g as negative.
Velocity-Time Graph
| Shape of v-t graph | Type of motion |
|---|---|
| Horizontal line | Uniform velocity (a = 0) |
| Straight inclined line (positive slope) | Uniform acceleration |
| Straight inclined line (negative slope) | Uniform deceleration |
| Curve | Non-uniform acceleration |
- Slope of v-t graph = acceleration.
- Area under v-t graph = displacement.
Position-Time Graph
- Slope of s-t graph = velocity.
- Horizontal line → object at rest.
- Straight inclined line → uniform velocity.
- Parabola → uniform acceleration.
- Steeper slope means greater velocity.
Distance vs Displacement
- Distance — total path length traversed; scalar; never negative.
- Displacement — straight-line vector from initial to final position; can be zero or negative.
- For a circular lap returning to start: distance = 2πr, displacement = 0.
- Speed = distance/time (scalar); velocity = displacement/time (vector).
CDS/OTA PYQ Examples
Q: Slope of a velocity-time graph gives:
(a) Displacement (b) Acceleration (c) Speed (d) Distance
Answer: (b) Acceleration.
Q: Area under a velocity-time graph gives:
(a) Acceleration (b) Displacement (c) Force (d) Velocity
Answer: (b) Displacement.
Q: A body falling freely from rest covers in 2 s (g = 10 m/s²):
(a) 10 m (b) 20 m (c) 30 m (d) 40 m
Answer: (b) 20 m. s = ½gt² = ½·10·4 = 20 m.
Q: For a body moving with uniform velocity, the v-t graph is:
(a) Inclined line (b) Horizontal line (c) Parabola (d) Vertical line
Answer: (b) Horizontal line — velocity constant.
Drill Velocity-Time Graphs and Displacement for CDS/OTA
CDS/OTA-pattern items on Velocity-Time Graphs and Displacement with answer keys and explanations.
Start Free Mock TestFrequently Asked Questions
Can distance be less than displacement?
No. Displacement ≤ distance always. They are equal only for motion in a straight line without reversal.
Why is a parabolic s-t graph an indicator of uniform acceleration?
Because s = ut + ½at² is quadratic in t — a parabola when plotted.
If velocity is zero, can acceleration be non-zero?
Yes — at the highest point of a vertically thrown ball, velocity is zero but g is still acting downward.