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Curvilinear motion is analyzed in three coordinate systems: (1) Cartesian — x,y components with constant unit vectors; (2) normal-tangential (n-t) — tangential direction along velocity with aₜ = dv/dt, normal direction toward center of curvature with aₙ = v²/ρ; (3) polar (r, θ) — with aᵣ = r̈ − rθ̇² and aθ = rθ̈ + 2ṙθ̇. The optimal coordinate system depends on the geometry and given information — circular paths favor n-t, problems stated in terms of angle favor polar.
Identify the most natural coordinate system for the problem geometry. Practice converting between systems. For n-t coordinates, always identify the center of curvature to establish the normal direction.
Rectilinear kinematics describes motion along a straight line, where velocity and acceleration always point in the same fixed direction. Curvilinear kinematics generalizes this to arbitrary curved paths, where the direction of motion changes continuously. The key challenge is that acceleration now has two components: one that changes the particle's speed and one that changes its direction.
The choice of coordinate system is not arbitrary — it should match the geometry of the problem. In Cartesian coordinates (x, y), the unit vectors are fixed, so velocity and acceleration simply decompose into two independent scalar equations. This works perfectly for projectile motion, where the x and y components decouple. For motion along a curved path of known shape, normal-tangential (n-t) coordinates are more natural: the tangential direction eₜ aligns with the velocity vector, and the normal direction eₙ points inward toward the center of curvature. The tangential acceleration aₜ = dv/dt governs how fast the speed changes; the normal acceleration aₙ = v²/ρ governs the rate of turning, where ρ is the radius of curvature at that point. A critical detail: even at constant speed, aₙ is nonzero — the particle is still accelerating because its direction is changing.
Polar coordinates (r, θ) are the coordinate system of choice when a problem is stated in terms of the distance from a fixed point and the angle swept. The radial acceleration aᵣ = r̈ − rθ̇² includes the familiar centripetal term −rθ̇² (which pulls inward for circular motion), and the transverse acceleration aθ = rθ̈ + 2ṙθ̇ includes the Coriolis term 2ṙθ̇. The Coriolis term surprises many students because it appears even when r is constant — it arises from the rotation of the coordinate frame itself, not from any physical force. Its appearance is purely a consequence of differentiating in a rotating reference system.
The practical skill is recognizing which system to use. If the path is described geometrically and you need to relate forces to the curvature of the path, use n-t coordinates. If the problem gives r(t) and θ(t) or describes motion in terms of angles from a pivot, use polar. If components separate naturally into horizontal and vertical, Cartesian is cleanest. Translating the same motion between systems is a useful exercise that sharpens understanding of what each coordinate system is actually measuring.