position velocity acceleration explained
A single particle moves along a straight line. Follow it over time and it crosses some stretches quickly and others so slowly that it almost stops, because the distance it covers in 1 s is not the same everywhere. The quantity that measures how fast position changes is velocity, and when velocity itself changes from moment to moment, how fast it changes needs a measure of its own: acceleration. Going from position to velocity, or from velocity to acceleration, is always the same step, asking how fast something changes, and that step is differentiation. Run it backward, piling up changes by integration, and you can climb back from acceleration to velocity and from velocity to position. All of particle kinematics rests on these two moves.
Drag the time t (seconds) with the slider and the gold particle slides along the track, with its current position x (meters) shown above it and in the readout. The faint rings mark where the particle was at equal time intervals: crowded at both ends, spread out in the middle. Crowded rings mean slow travel and spread rings mean fast, so this particle starts slowly, is fastest midway, and slows again at the end. Each time fixes exactly one position, and that relationship, x(t), holds the whole motion. The next scenes dig into how fast this position changes.
Now spread the same motion out as a graph, with time t on the horizontal axis and position x on the vertical; the green curve is the motion from the previous scene. Move the time with the slider and a blue tangent rides along with the point on the curve. Its slope is the velocity, v = dx/dt: how much the position changes over a very short time. In the steep middle the slope is large and the particle is fast; at the flat ends the slope is close to 0 and it has almost stopped. Velocity is the steepness of the slope, not the height of the curve, so a particle can be far from its start and momentarily at rest, or close to the start and moving fast.
The same move works one level up. This graph puts velocity v on the vertical axis; the blue curve shows how velocity changes, and the slope of the orange tangent at the point is the acceleration, a = dv/dt: how fast velocity changes. Just after the start the velocity is rising, so the slope and the acceleration are positive. At the peak of the velocity curve the tangent is level, the slope is 0, and at that instant the acceleration is 0 too. After that the velocity falls and the slope goes negative, which is deceleration. The key point is that differentiating position gives velocity, and differentiating velocity the same way gives acceleration.
Stack the three graphs and their relationship jumps out: position x on top, velocity v in the middle, acceleration a at the bottom. Switch the type of motion with the buttons and compare how the three curves fit together. In uniform motion, position is a straight slope, velocity a flat line, and acceleration 0. In uniformly accelerated motion, position is a curved parabola, velocity a rising straight line, and acceleration a flat line; when the acceleration itself varies, all three curves bend. Either way, the slope of each graph is the height of the one below it: where position is steep, velocity is high, and where velocity is steep, acceleration is large.
If differentiation takes you one way, you should be able to come back the other way. Return to the velocity graph and drag the time t with the slider: the area under the curve from 0 up to now fills in blue, and that area is the displacement so far. Velocity times time is distance, so stacking up v, the velocity at every moment, over time gives the total displacement. That is integration: where differentiation pulled a slope out of a curve, integration gathers up the area beneath it. Drag all the way to the end and the area matches the distance from start to finish exactly. So accumulating acceleration recovers velocity, and accumulating velocity recovers position.