Parametric, Polar & Vector Motion Lab

Trace parametric curves, shade polar regions, and decompose vector-valued motion into velocity and acceleration. Switch between three modes to see how different coordinate systems reveal different properties of curves and motion.

Guided Experiment: Parametric Motion Analysis

How does the velocity vector change along a cycloid? Is the speed constant?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

Results

Position(1.0000, 0.0000)
dx/dt0.0000
dy/dt1.0000
dy/dx
Speed |v|1.0000
Arc Length0.0000

Graph

Velocity

Data Table

(0 rows)
#TrialModePresett / θPositionDerivative / SlopeSpeed / Area / κ
0 / 500
0 / 500
0 / 500

Reference Guide

Parametric Derivatives

For a curve defined by x(t) and y(t), the slope is found using the chain rule.

dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

Arc length is the integral of the speed function.

L=∫ab(dxdt)2+(dydt)2 dtL = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt

Polar Area

The area enclosed by a polar curve between two angles is given by the following integral.

A=12∫αβr2 dθA = \frac{1}{2}\int_{\alpha}^{\beta} r^2\, d\theta

This formula sweeps out infinitesimal triangular sectors from the origin. The factor of 1/2 comes from the area of each triangle with base r and infinitesimal angle dθ.

Polar Slope

The slope of a polar curve in Cartesian coordinates requires converting derivatives.

dydx=drdθsin⁡θ+rcos⁡θdrdθcos⁡θ−rsin⁡θ\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}

This comes from x = r cos θ and y = r sin θ with the product rule.

Vector Calculus

For a vector-valued function r(t), velocity and acceleration decompose into tangential and normal components.

κ=∣v×a∣∣v∣3,aT=v⋅a∣v∣,aN=∣a∣2−aT2\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3}, \quad a_T = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|}, \quad a_N = \sqrt{|\mathbf{a}|^2 - a_T^2}

Curvature κ measures how quickly the direction of velocity changes. Tangential acceleration changes the speed, while normal acceleration changes the direction.