What do the slope of x–t and the area under v–t mean?

The slope of the tangent to an x–t graph is the instantaneous velocity: a straight line means constant velocity, a parabola means constant acceleration, opening upward when a > 0. The slope of v–t is the acceleration, and the signed area between the v–t line and the time axis is the displacement, positive above the axis and negative below. The area under a–t is the change in velocity.

What is the difference between displacement and distance?

Displacement is final minus initial position, the signed area under v–t. Distance is the total path length, the area under |v|. They are equal as long as the velocity keeps the same sign; once the object turns back (the v–t line crosses the axis) the distance exceeds the size of the displacement. Both are shown live.

Why is the x–t graph of free fall a parabola?

In free fall the acceleration is constant at −g, so velocity falls linearly with time (a sloping straight v–t line) and position x = x₀ − ½gt² is quadratic in time. The x–t graph is therefore a downward-opening parabola whose tangent gets steeper as the object speeds up.

Why is the bouncing ball only approximate?

A real bounce is a very large force acting for a very short time. Here each impact is modelled as a 0.02 s segment of large positive acceleration that reverses the velocity (restitution 0.75), with free fall at −g in between. That gives narrow, tall spikes on the a–t graph and a saw-tooth v–t graph.

How is the matching game scored?

Your curve is recorded every frame for 10 seconds and compared with the target at the same times; the root-mean-square error gives a score of 100·e^(−RMS/0.6). When matching x–t you control the velocity, so you must make the slope follow the target; matching v–t you control the acceleration, which is harder.

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📉Motion Graphs - Position, Velocity & Acceleration vs Time, Graph Matching

Split a straight-line motion into segments of constant acceleration and watch the object move along a track while the x–t, v–t and a–t graphs are drawn in sync. At the cursor you see the tangent to x–t (its slope is the velocity) and the shaded area under v–t (the displacement). Two practice modes are included: a matching game where you steer the velocity in real time to follow a target curve, and translation exercises that reveal the other two graphs step by step.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use Motion Graphs (x-t, v-t, a-t)

1Pick a preset (free fall, braking car, elevator, bouncing ball) or edit each segment’s duration and acceleration in the table
2Drag the dots on the v–t graph to change velocities, then drag the cursor or press Play
3In the matching game, steer the velocity (or acceleration) with the slider so your curve follows the target, and get a score
4In translation practice, sketch the missing graphs yourself, then reveal the answer and the working for each segment

FAQ

What do the slope of x–t and the area under v–t mean?: The slope of the tangent to an x–t graph is the instantaneous velocity: a straight line means constant velocity, a parabola means constant acceleration, opening upward when a > 0. The slope of v–t is the acceleration, and the signed area between the v–t line and the time axis is the displacement, positive above the axis and negative below. The area under a–t is the change in velocity.
What is the difference between displacement and distance?: Displacement is final minus initial position, the signed area under v–t. Distance is the total path length, the area under |v|. They are equal as long as the velocity keeps the same sign; once the object turns back (the v–t line crosses the axis) the distance exceeds the size of the displacement. Both are shown live.
Why is the x–t graph of free fall a parabola?: In free fall the acceleration is constant at −g, so velocity falls linearly with time (a sloping straight v–t line) and position x = x₀ − ½gt² is quadratic in time. The x–t graph is therefore a downward-opening parabola whose tangent gets steeper as the object speeds up.
Why is the bouncing ball only approximate?: A real bounce is a very large force acting for a very short time. Here each impact is modelled as a 0.02 s segment of large positive acceleration that reverses the velocity (restitution 0.75), with free fall at −g in between. That gives narrow, tall spikes on the a–t graph and a saw-tooth v–t graph.
How is the matching game scored?: Your curve is recorded every frame for 10 seconds and compared with the target at the same times; the root-mean-square error gives a score of 100·e^(−RMS/0.6). When matching x–t you control the velocity, so you must make the slope follow the target; matching v–t you control the acceleration, which is harder.

Features

Synced x–t, v–t and a–t graphs with tangent and signed area shading at the cursor
Draggable v–t knots and an editable table of durations and accelerations
Track animation with velocity and acceleration arrows and strobe dots every 0.5 s, including vertical motion
Live displacement and distance travelled (different when the velocity changes sign)
Graph-matching game with two target types and two difficulty levels, scored by RMS error
Translation practice from x–t, v–t or a–t with slope and area working for every segment

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