Position, Velocity & Acceleration vs. Time Graphs
Slope, area under the curve, direction, and worked motion examples
Quick Reference: Slope and Area
| Graph | Slope gives | Area under curve gives |
|---|---|---|
| Position–time (x-t) | Velocity | Not a standard physical quantity (rarely used) |
| Velocity–time (v-t) | Acceleration | Displacement (Δx) |
| Acceleration–time (a-t) | Rate of change of acceleration (rarely tested) | Change in velocity (Δv) |
⚠ Watch out:Area below the horizontal axis counts as negative. If a v-t graph dips below the axis, subtract that area instead of adding it when finding total displacement.
1. Position vs. Time Graphs
Slope of an x-t graph = velocity (v = ΔxΔt). A steeper line means a faster speed. A straight (linear) section means constant velocity; a curved section means the velocity is changing, so the object is accelerating. Concave up (curving upward) means increasing/more-positive velocity; concave down means decreasing/more-negative velocity.
Positive slope: moving in the + direction
Negative slope: moving in the - direction
Zero slope: at rest (not moving)
Peak/valley point: velocity = 0 for an instant (turning around)
Position vs. Time Examples
| Motion | Graph | Slope | Value / Sign | Details |
|---|---|---|---|---|
| At rest, away from the origin | Zero (flat) | Velocity = 0 | Acceleration = 0 Not moving | |
| Constant positive velocity | Constant, positive | Velocity is constant and positive | Acceleration = 0 Moving in the + direction at steady speed | |
| Constant negative velocity | Constant, negative | Velocity is constant and negative | Acceleration = 0 Moving in the - direction at steady speed | |
| Speeding up, + direction | Positive, getting steeper | Velocity is positive and increasing | Acceleration is positive Moving + and speeding up (curve bends upward) | |
| Slowing down, + direction | Positive, getting flatter | Velocity is positive and decreasing | Acceleration is negative Moving + and slowing down (curve levels off) | |
| Speeding up, - direction | Negative, getting steeper | Velocity is negative and its magnitude is increasing | Acceleration is negative Moving - and speeding up (curve bends downward) | |
| Slowing down, - direction | Negative, getting flatter | Velocity is negative and its magnitude is decreasing | Acceleration is positive Moving - and slowing down (curve levels off) | |
| Turning around (changes direction) | Positive, then zero at the peak, then negative | Velocity goes from + to 0 to - | Acceleration is negative the whole time (constant, if the curve is a parabola) e.g. a ball tossed straight up: rises, stops for an instant, falls back |
2. Velocity vs. Time Graphs
Slope of a v-t graph = acceleration (a = ΔvΔt). The signed area between the line and the t-axis = displacement (Δx). Area above the axis is positive displacement; area below the axis is negative displacement.
Same sign for a and v (both + or both -): object is speeding up
Opposite signs for a and v: object is slowing down
Line above axis: moving in the + direction
Line below axis: moving in the - direction
Velocity vs. Time Examples
| Motion | Graph | Slope | Value / Sign | Details |
|---|---|---|---|---|
| At rest | Zero (flat, on the axis) | v = 0 | a = 0. Area under the curve = 0 (no displacement) Not moving | |
| Constant positive velocity | Zero (flat, above the axis) | v is constant and positive | a = 0. Shaded area (rectangle) = positive displacement Moving steadily in the + direction | |
| Constant negative velocity | Zero (flat, below the axis) | v is constant and negative | a = 0. Shaded area (below axis) = negative displacement Moving steadily in the - direction | |
| Speeding up, + direction | Positive (line rises, above the axis) | v is positive and increasing | a is positive (same sign as v). Area (triangle) = + displacement Moving + and gaining speed | |
| Slowing down, + direction | Negative (line falls, stays above the axis) | v is positive and decreasing | a is negative (opposite sign of v). Area (triangle) = + displacement Moving + but losing speed, e.g. braking | |
| Speeding up, - direction | Negative (line falls, below the axis) | v is negative and its magnitude is increasing | a is negative (same sign as v). Area (below axis) = - displacement Moving - and gaining speed | |
| Slowing down, - direction | Positive (line rises, stays below the axis) | v is negative and its magnitude is decreasing | a is positive (opposite sign of v). Area (below axis) = - displacement Moving - but losing speed, coming to a stop | |
| Changing direction (crosses the axis) | Constant, negative (straight line through the axis) | v starts positive, passes through 0, becomes negative | a is constant and negative. Area above axis (+) partly cancels area below axis (-) e.g. ball thrown straight up under gravity: rises then falls |
Example: Find the displacement from a v-t graph that is a straight line from v = 0 at t = 0 to v = 8 m/s at t = 4 s.
- The region under the line is a triangle: base = 4 s, height = 8 m/s
- Area = 12(base)(height) = 12(4)(8)
- Answer: Δx = 16 m
3. Acceleration vs. Time Graphs
The signed area between an a-t graph and the t-axis = change in velocity (Δv = aΔt). The slope of an a-t graph (how quickly acceleration itself is changing) is rarely tested in introductory physics, but a flat a-t line means constant acceleration, which is the case covered by the kinematic (SUVAT) equations.
a = 0 (on the axis): velocity is constant (uniform motion or rest)
a above axis (+): velocity is increasing (becoming more positive)
a below axis (-): velocity is decreasing (becoming more negative)
Step change: a switches suddenly, e.g. a force turns on or off
Acceleration vs. Time Examples
| Motion | Graph | Slope | Value / Sign | Details |
|---|---|---|---|---|
| No acceleration | Zero | a = 0 | Velocity is constant (or the object is at rest). Area under curve = 0 = no change in v Object moves at steady velocity or stays still | |
| Constant positive acceleration | Zero (flat line above the axis) | a is constant and positive | Shaded area (rectangle) = the increase in velocity, Δv = aΔt Velocity is increasing in the + direction (or decreasing in magnitude if moving -) | |
| Constant negative acceleration (braking / gravity) | Zero (flat line below the axis) | a is constant and negative | Shaded area (below axis) = the decrease in velocity Velocity is decreasing, e.g. free fall (a = -9.8 m/s²) or a car braking | |
| Acceleration increasing over time | Positive and increasing (curve steepens) | a is positive and growing | Velocity is changing faster and faster (a rocket burning more fuel) The push/force on the object is getting stronger over time | |
| Acceleration suddenly changes (step) | Zero, then an abrupt jump, then zero again | a switches instantly from one constant value to another | e.g. engine cuts off (a drops from positive to negative/zero) Two separate constant-acceleration intervals joined together | |
| Acceleration decreasing toward zero | Positive, decreasing toward zero | a is positive but shrinking | Object approaches a maximum (terminal) velocity as a → 0 e.g. a skydiver speeding up but leveling off near terminal velocity |
⚠ Watch out:A positive acceleration does not always mean an object is speeding up — it only means speeding up when velocity is also positive. If velocity is negative, a positive acceleration means the object is slowing down.
4. Matching All Three Graphs to the Same Motion
Since each graph is derived from the one before it (slope of x-t = v-t, slope of v-t = a-t), the same motion looks different on each graph. Practice translating a description into all three.
Same Motion, Three Graphs
| Description | Position vs. Time | Velocity vs. Time | Acceleration vs. Time |
|---|---|---|---|
| Stationary object | |||
| Constant velocity (+) | |||
| Constant positive acceleration (starts at rest) | |||
| Ball tossed straight up | |||
| Braking to a stop (from + velocity) |
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