Use calculus in kinematics for motion in a straight line: v = dr/dt, a = dv/dt = d²r/dt²; r = ∫v dt, v = ∫a dt; extend to 2 dimensions using vectors

Calculus in kinematics allows you to model motion using differentiation and integration. For motion in a straight line, velocity v is the rate of change of displacement r with respect to time t: v = dr/dt. Acceleration a is the rate of change of velocity: a = dv/dt = d²r/dt². Conversely, displacement is the integral of velocity: r = ∫v dt, and velocity is the integral of acceleration: v = ∫a dt. This topic is central to Edexcel A-Level Mathematics (Mechanics) and appears in both pure and applied contexts, often in exam questions that require you to derive equations of motion from given functions.

Extending to two dimensions involves using vectors. Displacement, velocity, and acceleration become vector quantities, typically expressed in terms of i and j components. For example, if r = x(t)i + y(t)j, then v = dr/dt = (dx/dt)i + (dy/dt)j, and a = dv/dt = (d²x/dt²)i + (d²y/dt²)j. Integration works component-wise: r = ∫v dt = (∫v_x dt)i + (∫v_y dt)j. This extension is crucial for problems involving projectile motion or any motion in a plane, and it builds on your understanding of vectors from pure mathematics.

Mastering this topic is essential for solving real-world problems in physics and engineering, and it frequently appears in exam questions that test both your calculus skills and your ability to interpret vector results. You'll need to be comfortable with differentiating and integrating polynomials, trigonometric functions, and exponentials, as well as applying initial conditions to find constants of integration. The topic also links to SUVAT equations, which are special cases when acceleration is constant.