Introduction
Where statics deals with bodies at rest, dynamics deals with bodies in motion under the action of forces — applying and extending the same Newtonian framework introduced in Physics Chapter I, but with a mathematics-syllabus emphasis on setting up and solving the resulting equations rigorously.
1. Particle motion in straight and curved paths
For motion along a straight line, position, velocity, and acceleration are related by simple derivatives (v = dx/dt, a = dv/dt) and, for constant acceleration, by the standard “equations of motion” (v = u + at, s = ut + ½at², v² = u² + 2as). For motion along a curved path, velocity is always tangent to the path, while acceleration generally has two components: a tangential component (changing speed) and a normal (centripetal) component (changing direction, directed toward the centre of curvature, magnitude v²/r for circular motion) — recognising and separating these two components correctly is essential for any curved-path problem.
2. Newton’s laws and equations of motion
Applying F = ma to a dynamics problem means identifying every force acting on the particle (often varying with position, velocity, or time), writing Newton’s second law as a differential equation, and solving it (frequently connecting directly to the ordinary differential equations of Chapter IV) subject to given initial conditions (starting position and velocity).
3. Work and energy
The work–energy theorem states that the net work done on a particle equals its change in kinetic energy, W = ΔKE — this is often a faster route to a solution than directly integrating the equation of motion, particularly when only speeds (not detailed time-dependence) are required. For conservative forces (like gravity, or an ideal spring), work done is independent of path and can be expressed via a potential energy function, allowing the full mechanical energy conservation principle (KE + PE = constant) to be applied.
4. Impulse and momentum
Impulse is the time-integral of force, J = ∫F dt, and by Newton’s second law this equals the resulting change in momentum, J = Δp — the impulse–momentum theorem. This is especially useful for collision and impact problems, where the exact force during a very short contact time is unknown or irrelevant, but the resulting change in momentum can still be calculated or measured.
5. Projectiles, circular motion, and related problems
Projectile motion treats horizontal and vertical motion as independent: horizontal velocity remains constant (no horizontal force, ignoring air resistance), while vertical motion follows constant-acceleration equations under gravity — combining the two independently, then eliminating time between them, gives the trajectory’s shape (a parabola) and key quantities like range, maximum height, and time of flight. Circular motion (uniform or non-uniform) requires identifying the centripetal force (whatever physical force — tension, gravity, friction, normal force — happens to supply the necessary v²/r acceleration toward the centre in a given problem) and, for non-uniform circular motion, a separate tangential force component changing the speed.
Key Takeaways
- Curved-path motion requires separating acceleration into tangential (speed-changing) and normal/centripetal (direction-changing) components.
- Newton’s second law, work–energy, and impulse–momentum are three different but consistent routes to solving a dynamics problem; choosing the most efficient one for the specific quantities asked for is a key exam skill.
- Conservative forces permit full mechanical energy conservation (KE + PE = constant), simplifying problems that would otherwise require solving a full equation of motion.
- Projectile motion decomposes cleanly into independent horizontal (constant velocity) and vertical (constant acceleration) components; circular motion requires correctly identifying which physical force supplies the necessary centripetal force.