Motion Graphs Simulator

What Is the Motion Graphs Simulator?

Read the Same Trip Two Ways

One journey, two graphs, two very different-looking stories. Set the Initial velocity u and the Acceleration a, press Play, and the dot slides along its track while the position-time and velocity-time graphs draw side by side. The pictures look nothing alike, yet they describe the exact same motion. What ties them together is one idea: the slope of each graph hands you the next quantity down.

On the position-time graph, the slope at any instant is the velocity. Give the dot a steady a and that curve bends into a parabola, because its steepness keeps growing as the dot speeds up. A straight, sloped position line would instead mean constant velocity and zero acceleration. On the velocity-time graph the slope is the acceleration, so a constant a draws a straight sloped line, and the area beneath it equals the displacement, following s = u·t + ½·a·t². Here is the trap: read the slope, never the height. A tall point on one graph is not the value on the other.

Watch how a curved position-time graph and a straight velocity line report the same run of v = u + a·t. Nudge a (in m/s²) toward zero and the velocity-time line flattens to horizontal while the position-time graph straightens into a sloped line — a reminder that a straight, tilted position graph still means steady motion. To rebuild the acceleration behind these slopes, open the acceleration lab, or step through every term in the SUVAT lab. Then pin down the exact numbers with the acceleration calculator and browse the full collection of physics sims.

Where the ideal motion graph breaks down

Every graph here rests on one assumption you can see in the acceleration panel: that line is horizontal, always. That is a choice, not a fact about the world. A car's engine pushes less hard the faster it goes, a braking car's deceleration shifts as weight transfers onto the front tyres, and a falling body meets rising air resistance. Once a stops being constant the two reading rules survive — slope gives the quantity below, area gives the one above — but s = u·t + ½·a·t² does not. The acceleration here also switches on at t = 0 as a vertical step, which would take an infinite rate of change of force; real lifts and trains deliberately ramp acceleration in and out so passengers do not stumble.

Set u to 10.0 m/s and a to −5.0 m/s², press Play, and read the position at the end: −12.5 m. The dot did not travel 12.5 m. It ran forward, slowed, turned round and came back past its starting line, and the graph reports only where it finished. On the velocity-time panel the line drops through zero and the area below the axis subtracts from the area above. A car's odometer would show a far larger number than any position readout here, and that gap is the whole difference between displacement and distance.

Finally, these curves are exact and real data never is. A school motion sensor samples position perhaps twenty times a second with a centimetre or so of scatter, and taking a slope from sampled data amplifies that scatter: a wobble invisible on the position trace makes the velocity trace visibly rough, and differentiating a second time turns the acceleration trace into noise. That is why careful measurements run the other way, integrating up from an accelerometer rather than differentiating down twice from position.

Where this is actually used

Stopping distance is the area of one triangle. Set u to 10.0 m/s and a to −2.0 m/s²: the velocity readout reaches 0.0 m/s at exactly 5.0 s and the position readout stops at 25.0 m, the triangle of height 10 and base 5, halved. This is why speed limits bite harder than they look. Double the entry speed at the same braking rate and both the height and the base of that triangle double, so the area quadruples — four times the tarmac, not twice. The sliders stop at 10.0 m/s; the geometry does not.

A take-off run is the same triangle read forwards. Leave u at 0.0 m/s with a at 2.0 m/s² and after 5.0 s the readouts give 10.0 m/s and 25.0 m — exactly half the 50.0 m those same sliders cover at a steady 10.0 m/s with a set to zero. Starting from rest costs you half your distance for nothing. An airliner leaves the ground near 80 m/s, and held to that same 2.0 m/s² it needs 40 s and 1,600 m of ground roll, which is why major runways are three kilometres long rather than one.

Your phone reads these panels from the bottom up. Its accelerometer measures a; integrating once gives velocity and twice gives position, the same two area steps run in reverse. It works for a few seconds and drifts badly after that, because a bias of even 0.01 m/s² feeds through the ½·a·t² term and becomes 18 m of phantom displacement after one minute. That is why a phone falls back on GPS the moment it can, and why aircraft inertial units are rated by how many metres per hour they wander.