Velocity–Time Graph Calculator
A motion in three phases — accelerate, cruize, brake — worked as a velocity–time graph: the speed at each corner, the distance under each segment, the total displacement and the average velocity.
On a velocity–time graph the slope of each segment is the acceleration and the area under it is the displacement.
How the velocity–time graph calculator works
On a velocity–time graph the slope of each segment is the acceleration and the area under it is the displacement. A phase of constant acceleration is a trapezium: its area is the average of the start and end speeds times the time. Add the three areas and you have the total distance, whatever the shape.
This is the graph every train, lift and robot arm follows — and reading it is faster than solving the equations of motion three times.
Formula: v = v₀ + a t; s = (v₀ + v) t / 2 for each phase; total = Σ s
Worked examples
| Inputs | Total displacement (m) | Note |
|---|---|---|
| A train between stations | 1,872 | stops exactly |
| Braking too short | 1,845 | still moving at the end |
| Braking too long | 1,811.25 | reverses |
FAQFrequently asked questions
How do I read a velocity–time graph?
Slope is acceleration; area under the line is displacement. A flat segment is constant speed, a rizing one acceleration, a falling one braking.
Why is the area a trapezium?
Because velocity changes linearly under constant acceleration, so the region under it has straight sides. Average of the two speeds times the time is its area.
What is the difference between displacement and distance?
Displacement is net change of position and can be reduced by reversing; distance counts every meter travelled. They differ only when the motion reverses.
How do I make it stop exactly?
Braking time must equal cruize speed over braking deceleration: 24 m/s at 1.5 m/s² is 16 seconds. Shorter and it is still moving; longer and it reverses.
Can I model more phases?
Chain them: the final velocity of one run is the initial velocity of the next, and displacements add.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Institute of Standards and Technology — the US measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.