PhysicsDifficulty 1-2

Motion

The change in an object's position over time relative to a reference frame. Kinematics describes motion using position, velocity (the rate of change of position) and acceleration (the rate of change of velocity), without yet asking what causes it.

Kinematics in Calculus Form

The SUVAT equations are an algebraic shortcut for the constant-acceleration case. The more general definitions treat velocity and acceleration as derivatives of position with respect to time.

v(t) = \dfrac{dx}{dt}, \qquad a(t) = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}

Velocity as the first time-derivative of position; acceleration as the second.

For constant acceleration a, integrating a(t) = a once with respect to time (using initial velocity u) gives v(t) = u + at; integrating again (using initial position x0) gives x(t) = x0 + ut + ½at^2 — the same SUVAT results, now derived rather than stated.

v(t) = u + \int_0^t a\,dt', \qquad x(t) = x_0 + \int_0^t v(t')\,dt'

General velocity and position as integrals of acceleration and velocity respectively.

Info: This derivative/integral relationship generalises directly to time-varying, non-constant acceleration, where the algebraic SUVAT equations no longer apply but the calculus definitions of v and a still do.

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Sources and methodology

Velocity is the rate of change of position with time and is a vector quantity, having both magnitude and direction.
- Fundamentals of Physics (Halliday, Resnick & Walker)
Average velocity equals the change in position divided by the change in time (v = Δx/Δt).
- Fundamentals of Physics (Halliday, Resnick & Walker)
Acceleration is the rate of change of velocity with time (a = Δv/Δt).
- Fundamentals of Physics (Halliday, Resnick & Walker)
An object moving at constant velocity requires zero net force to continue moving — steady motion does not require a continuous applied force.
- Fundamentals of Physics (Halliday, Resnick & Walker)

Content status: published, last reviewed 30 September 2026.