Parametric curves – velocity, dy/dx and arc length

What it shows

A parametric curve gives both coordinates as functions of a parameter t, so x = x(t) and y = y(t) describe a path traced in a definite direction. Reading t as time turns the curve into motion: the velocity vector is (x′, y′), the speed is √(x′² + y′²) and the acceleration is (x″, y″). The gradient is dy/dx = y′/x′, undefined where x′ = 0, and eliminating t gives the Cartesian equation. The arc length is the integral of the speed. Derivatives here come from central differences and the integral from Simpson's rule, so values are close approximations.

How to use

Choose a Curve, then drag the red point P or the Parameter t slider, or press Play. Change a and b to reshape the curve. Use Velocity and acceleration, Tangent line and Polyline to show each feature, and increase Polyline segments n to see Lₙ approach L in the table. To try another curve, type x(t), y(t) and the t-range: any edit switches to Your own x(t), y(t).

Parameters you can change

• Curve Line, Circle, Ellipse, Projectile (g = 9.81 m/s²), Lissajous curve, Cycloid, Your own x(t), y(t)
• Formula for x(t) (own curve)
• Formula for y(t) (own curve)
• Start value t₀ (own curve)
• End value t₁ (own curve)
• Starting position of t (percentage of the t-range) 0–100 %
• Polyline segments n 1–64
• Show velocity and acceleration vectors
• Show the tangent line
• Show the polyline

Questions to explore

1. On the circle, why is the acceleration always perpendicular to the velocity?
2. Where on the ellipse is the tangent vertical, and what happens to dy/dx there?
3. How many polyline segments make Lₙ less than 1 % shorter than the cycloid's arc length?