Position Time Graph to Velocity Time Graph: How to Convert Between Both Graphs
The slope of a position time graph at any point equals the velocity at that moment. To convert a position time graph to a velocity time graph, calculate the slope of each section and plot those values on the velocity time graph’s y-axis at the corresponding times. For the equations behind constant-acceleration graph problems, see the SUVAT equations guide.
For a supporting reference, Khan Academy’s position vs. time graphs guide explains how slope on a position-time graph represents velocity.
What Is the Relationship Between a Position Time Graph and a Velocity Time Graph?
The slope of a position time graph gives velocity. The y-axis value of a velocity time graph gives position change per unit time. These 2 graphs describe the same motion from different perspectives.
On a position time graph, the y-axis shows position in metres (m) and the x-axis shows time in seconds (s). On a velocity time graph, the y-axis shows velocity in metres per second (m/s) and the x-axis shows time in seconds (s).
How Do You Calculate Velocity from a Position Time Graph?
Velocity equals the slope of the position time graph, calculated using the formula: v = Δx / Δt, where Δx is the change in position and Δt is the change in time.
Select 2 points on the line, read their position and time values from the axes, and apply the formula:
v = (x₂ – x₁) / (t₂ – t₁)
For example: a position time graph shows position increasing from 0 m at t = 0 s to 15 m at t = 3 s.
v = (15 – 0) / (3 – 0) = 5 m/s
This value of 5 m/s becomes the y-axis value on the velocity time graph across that 3-second interval.
What Does Each Line Shape on a Position Time Graph Produce on a Velocity Time Graph?
There are 5 position time graph shapes, each producing a distinct line on the velocity time graph. The table below shows every conversion.
| Position Time Graph Line | Slope Value | Velocity Time Graph Line |
|---|---|---|
| Straight line, positive slope | Constant positive | Horizontal line above x-axis |
| Straight line, negative slope | Constant negative | Horizontal line below x-axis |
| Horizontal line (flat) | Zero | Horizontal line on x-axis (v = 0) |
| Curve, concave up (steepening) | Increasing positive | Straight line sloping upward |
| Curve, concave down (flattening) | Decreasing positive | Straight line sloping downward |
A straight line on a position time graph always produces a horizontal line on the velocity time graph. A curve on a position time graph always produces a sloped straight line on the velocity time graph, assuming uniform acceleration.
How Do You Convert a Position Time Graph to a Velocity Time Graph Step by Step?
Converting a position time graph to a velocity time graph follows 5 steps.
- Split the position time graph into separate sections. Each section is a segment where the line shape is consistent: straight, curved, or flat.
- Identify the slope of each section. Determine whether it is positive, negative, or zero, and whether it is constant or changing.
- Calculate the slope numerically for each straight-line section using v = Δx / Δt.
- Plot each slope value as a horizontal line on the velocity time graph across the same time interval.
- For curved sections, note whether the slope is increasing or decreasing and draw a sloped line on the velocity time graph in the corresponding direction.
How to Convert a Position Time Graph to a Velocity Time Graph: Worked Example
Consider a position time graph with 3 sections over 8 seconds.
Section 1: Constant Positive Slope (0 s to 3 s)
Position increases from 0 m to 12 m in 3 seconds.
Slope = (12 – 0) / (3 – 0) = 4 m/s
On the velocity time graph: draw a horizontal line at v = 4 m/s from t = 0 s to t = 3 s.
Section 2: Zero Slope (3 s to 5 s)
Position stays constant at 12 m. The line is flat.
Slope = 0 m/s
On the velocity time graph: draw a horizontal line at v = 0 m/s from t = 3 s to t = 5 s.
Section 3: Constant Negative Slope (5 s to 8 s)
Position decreases from 12 m to 0 m in 3 seconds.
Slope = (0 – 12) / (8 – 5) = -12 / 3 = -4 m/s
On the velocity time graph: draw a horizontal line at v = -4 m/s from t = 5 s to t = 8 s.
What Does a Curved Position Time Graph Look Like on a Velocity Time Graph?
A curved position time graph produces a sloped straight line on the velocity time graph, provided the acceleration is uniform. There are 2 types of curve.
- Concave up curve (slope increasing): the position time graph steepens over time. Velocity is increasing. On the velocity time graph, this produces a straight line with a positive slope.
- Concave down curve (slope decreasing): the position time graph flattens over time. Velocity is decreasing. On the velocity time graph, this produces a straight line with a negative slope.
To find velocity at a specific point on a curve, draw a tangent line at that point and calculate its slope. The slope of the tangent line on a position time graph represents the instantaneous velocity at that moment.
What Does a Parabola on a Position Time Graph Look Like on a Velocity Time Graph?
A parabola on a position time graph produces a straight diagonal line on the velocity time graph. This is because a parabola has a slope that increases or decreases at a constant rate, meaning acceleration is constant.
There are 2 cases.
- Upward parabola (concave up, increasing slope): the object accelerates in the positive direction. The velocity time graph shows a straight line with a positive slope, starting from the initial velocity at t = 0.
- Downward parabola (concave down, decreasing slope): the object decelerates. The velocity time graph shows a straight line with a negative slope.
For example, an object in free fall follows a parabolic position time graph. On the velocity time graph, this appears as a straight line with slope -9.8 m/s², representing constant gravitational acceleration.
What Are the 3 Rules for Converting Any Position Time Graph to a Velocity Time Graph?
There are 3 rules that apply to every conversion.
- Slope becomes y-value: the slope of the position time graph at any time equals the y-axis value (velocity) on the velocity time graph at that same time.
- Constant slope becomes flat line: any straight-line section on a position time graph becomes a horizontal line on the velocity time graph. The height of that horizontal line equals the slope of the position time graph section.
- Changing slope becomes a sloped line: any curved section on a position time graph becomes a sloped straight line on the velocity time graph. A steepening curve produces an upward slope. A flattening curve produces a downward slope.
What Is the Difference Between a Position Time Graph and a Velocity Time Graph?
A position time graph and a velocity time graph each reveal different information about the same motion. There are 4 key differences.
| Feature | Position Time Graph | Velocity Time Graph |
|---|---|---|
| Y-axis quantity | Position (m) | Velocity (m/s) |
| Slope represents | Velocity (m/s) | Acceleration (m/s²) |
| Flat horizontal line means | Object at rest | Constant velocity |
| Area under graph | Not directly useful | Displacement (m) |
What Are Common Errors When Converting from a Position Time Graph to a Velocity Time Graph?
There are 4 errors that occur frequently during conversion.
- Plotting position values instead of slope values: the y-axis of the velocity time graph shows slope, not position. Do not copy position values directly onto the velocity time graph.
- Missing sign changes: a slope changing from positive to negative on the position time graph means the object reversed direction. The velocity time graph line must cross the x-axis at that point.
- Ignoring turning points: a peak or trough on a position time graph means velocity = 0 at that moment. The velocity time graph must touch the x-axis at the same time.
- Confusing a curve with a straight line: a straight line on a position time graph produces a flat horizontal line on the velocity time graph, not a diagonal one. Only a curved position time graph produces a sloped line on the velocity time graph.
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