Velocity and Acceleration as Derivatives
Velocity is the time derivative of the position vector and acceleration the derivative of velocity, taken one component at a time. Motion whose acceleration changes needs no new formulas once this is the definition.
What a learner can do afterwards
- Differentiates a position function to get velocity and acceleration at any instant
- Handles two-dimensional motion by differentiating each component separately
- Says why the constant-acceleration equations stop applying as soon as acceleration depends on time
1 · Read
A speedometer needs more than distance: it needs to know how fast that distance is changing right now. That rate of change is velocity, the time derivative of position, and acceleration is the derivative of velocity in turn, making it the second derivative of position. Take s(t) equal to t cubed plus 2t. Then v(t) equals 3t squared plus 2, and a(t) equals 6t. At t equal to 2, velocity is 14 and acceleration is 12.
In two dimensions position is a vector with one component per axis. Differentiate each component. For r(t) equal to the pair t squared and 3t, velocity is the pair 2t and 3, and acceleration is the pair 2 and 0. At t equal to 2 velocity is the pair 4 and 3, whose length is 5. Speed is that length, always nonnegative, while velocity keeps its direction.
Velocity is the limit of displacement over time as the window shrinks to zero, which is what a speedometer reads. The sign of velocity carries the direction along the axis: positive one way, negative the other.
The constant acceleration equations stop applying the moment acceleration depends on time. With a(t) equal to 6t the acceleration keeps changing, so those formulas stay on the shelf. Differentiate or integrate instead: velocity is the derivative of position and acceleration the derivative of velocity, one component at a time.
Differentiate position for velocity and velocity for acceleration, work axis by axis, and drop the constant acceleration formulas once acceleration varies.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.