Introduction
Kinematics describes how objects move without asking why they move. In earlier work you read displacement, velocity, and acceleration from graphs. Calculus unlocks a more powerful approach: given any one of these quantities as a function of time, you can find the others exactly — not just from a straight-line segment on a graph, but for any motion expressible by a polynomial, trigonometric, or exponential rule.
This is a high-value topic on 9709 Mechanics papers. Questions routinely ask you to find the maximum speed, the time at which a particle is at rest, or the total distance travelled — all requiring calculus rather than graph-reading.
The calculus used is restricted to techniques from Pure Mathematics 1: differentiation and integration of polynomials (and simple extensions), so the algebra stays manageable.
Core Concept
Let denote time (seconds) and let a particle move in a straight line. Define:
- — displacement from a fixed origin (metres), a function of
- — velocity (m s), rate of change of displacement
- — acceleration (m s), rate of change of velocity
The calculus links between them form a two-way chain:
Differentiating moves you down the chain (displacement → velocity → acceleration).
Integrating moves you up the chain (acceleration → velocity → displacement), introducing a constant of integration each time that must be found from an initial or boundary condition.
A crucial distinction:
- Displacement is a signed quantity (positive or negative relative to the origin).
- Distance travelled counts all motion regardless of direction; if the particle reverses, you must split the integral at the turning point (where ).
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