JEE/NEET Physics · Motion in a Straight Line series · Part 3 of 6 · All parts →
- x-t graph: position against time — slope = velocity (flat = at rest)
- v-t graph: velocity against time — slope = acceleration; AREA under it = displacement
- a-t graph: acceleration against time — area = change in velocity
- Straight sloping line = constant velocity/acceleration; curve = changing
- Shapes tell stories: parabola in x-t means constant a; triangle area in v-t means uniform braking
Every journey is a story, and motion graphs are how physics writes it down. Learn to read the three graphs and most kinematics problems become pictures instead of algebra. Part 3 of the Motion in a Straight Line series.
- The x-t graph: where are you?
- The v-t graph: the most useful picture in mechanics
- The a-t graph
- Shape dictionary
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The x-t Graph: Where Are You?
Plot position against time. Slope = velocity: steep = fast, flat = standing still, downhill = moving backwards. A straight sloping line is uniform velocity; a curve means velocity is changing — a parabola is the signature of constant acceleration (the same s = ut + ½at² drawn as a picture).
The v-t Graph: The Most Useful Picture in Mechanics
Plot velocity against time. Two rules unlock everything: slope = acceleration (tilt up = speeding up, flat = constant velocity, tilt down = braking) and area under the graph = displacement (count area below the axis as negative — backwards motion).
The a-t Graph
Acceleration against time: its area = change in velocity. A flat non-zero a-t line is a steadily tilting v-t line — the graphs are each other’s slopes and areas, all the way up.
Shape Dictionary
| Graph shape | x-t means | v-t means |
|---|---|---|
| Horizontal flat line | at rest | constant velocity (a = 0) |
| Straight sloped line | uniform velocity | uniform acceleration |
| Parabola (curve) | uniform acceleration | changing acceleration |
| Line crossing zero | direction reversal | velocity sign flip (turnaround) |
| Triangle | — | braking to stop: area = stopping distance |
Solved Examples
Stage 1 area (triangle): ½ × 5 × 20 = 50 m.
Stage 2 (rectangle): 5 × 20 = 100 m.
Total = 150 m. Acceleration = slope = 20/5 = 4 m/s². ✔
Answer: 150 m
Curved x-t = changing velocity; upward parabola = slope increasing = accelerating in the + direction, starting from rest if it begins flat.
This is s = ½at² drawn as a picture. ✔
Answer: Uniform acceleration from rest
Area above axis: ½ × 4 × 10 = +20 m; area below: ½ × 4 × 6 = −12 m.
Net = +8 m, while distance = 32 m. The sign of the area did the direction bookkeeping automatically. ✔
Answer: +8 m (distance 32 m)
- Reading slope as value. On x-t, a HIGH line means ‘far away’, not ‘fast’ — speed is the slope, never the height.
- Forgetting area below the axis is negative in v-t — displacement must subtract backwards motion; distance doesn’t.
- Curve = acceleration, straight = constant. Any curvature in x-t means velocity is changing; there is no ‘straight but accelerating’.
- Confusing the graphs’ axes. x-t and v-t shapes look similar but mean different things — always check the axis labels first.
This Physics in Your Daily Life
- Fitness apps draw your v-t graph live — pace flat while cruising, a dip on hills, a spike at the sprint finish: slope and area exactly as this card.
- Flight data recorders (black boxes) reconstruct crashes from acceleration graphs — area gives velocity change, twice more area gives the crash story.
- Cricket ball speed replays — the graphics show a v-t story: release spike, slight decay through the air, brutal reversal at the bat.
- Traffic-engineering ‘speed profiles’ — city planners design roads from drivers’ v-t graphs: smooth slopes = safe junctions; sharp triangles = accident spots.
- EV regenerative braking displays — the energy recovered is literally the area under the deceleration part of your v-t graph.
An equation is the script of a film; a graph is the film itself. ‘The car braked, stopped a moment, reversed’ is one glance at a v-t line crossing zero — but three rounds of equation-solving if you insist on algebra. Graphs let your eyes do the physics.
Braking to a stop: v-t falls from 30 to 0 in 6 s. The triangle under it: ½ × 6 × 30 = 90 m of stopping distance — no equation needed, just area counting. Reverse for 3 s at 10: area −15. Net displacement: 75 m, distance 105 m. Count squares, get answers.
Three panels stacked vertically — x-t on top, v-t middle, a-t bottom — connected by arrows: each graph’s SLOPE gives the one below; each graph’s AREA gives the one above. One motion, three floors of the same building, with slope-stairs and area-lifts between them.
Practice set (answers hidden — try first)
(NEET-level) Flat line on a v-t graph means:
(JEE Main-level) v-t triangle 0→24 m/s in 8 s: displacement =
(NEET-level) x-t slope is negative and constant:
(Concept) An upward parabola on x-t indicates:
(JEE Main-level) v: +12 (3 s) then −6 (2 s), both triangles: |displacement| =
- x-t slope = velocity; flat = rest
- v-t slope = acceleration; area = displacement (signed!)
- a-t area = Δv
- curve = changing quantity; straight = constant
- reversal = graph crossing the axis
- 🔁 x-t: slope = v
- 🔁 v-t: slope = a, area = s (signed)
- 🔁 a-t: area = Δv
- 🧠 Chant: ‘slope goes DOWN the stack, area goes UP’.
- 🧠 v-t triangle = braking distance — count it, don’t derive it.
- 🏠 Daily: running-app pace curves = your personal v-t graph.
- 🏠 Daily: black boxes and EV regen displays run on graph areas.
Quick revision
- x-t graph: position against time — slope = velocity (flat = at rest)
- v-t graph: velocity against time — slope = acceleration; AREA under it = displacement
- a-t graph: acceleration against time — area = change in velocity
- Straight sloping line = constant velocity/acceleration; curve = changing
- Shapes tell stories: parabola in x-t means constant a; triangle area in v-t means uniform braking
- The x-t graph: where are you?
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