AP Calculus motion explorer

See position, velocity, and acceleration work together.

Build intuition for particle motion along a line. Watch the particle move, compare the three graphs, and connect slope, sign, speed, turning points, and concavity in one place.

Position tells you where the particle is.
Velocity tells direction and speed.
Acceleration tells how velocity changes.
Core idea
v = s′
Next link
a = v′ = s″
Motion test
sign of v

What to look for

When position rises, velocity is positive.

If the position graph is increasing, the particle is moving to the right. If it is decreasing, the particle is moving to the left.

When velocity crosses zero, direction can change.

That is where the particle may stop and turn around, if the sign of velocity changes.

Acceleration and velocity together control speeding up or slowing down.

Same sign means speed increases, opposite signs mean speed decreases.

Graph connections

Each graph shares the same time axis. A vertical guide follows the same moment t in all three.

Position Velocity Acceleration

Position graph

s(t) = t^3 - 6t^2 + 9t

Velocity graph

v(t) = slope of position

Acceleration graph

a(t) = slope of velocity

Particle moving on a line

Watch the particle move left and right as time changes. This is the same motion encoded in the graphs above.

t = 0.00
Position
0
Where the particle is on the line right now.
Velocity
0
Positive means moving right. Negative means moving left.
Acceleration
0
Shows whether velocity is increasing or decreasing.