---
title: "Polar-Coordinate Kinematics Interactive Lab | Sandbox Physics"
description: "How do you find the velocity and acceleration of a point on an arm that stretches while it turns?"
canonical_url: "https://sandboxphysics.com/labs/polar-kinematics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/polar-kinematics/"
last_modified: "2026-10-27"
---

M009 · Moving bases

# Polar-Coordinate Kinematics

**How do you find the velocity and acceleration of a point on an arm that stretches while it turns?** Velocity is the stretching speed along the arm plus the radius times the turning rate across it. Differentiating again, the turning basis adds two terms to the radial and angular accelerations: an inward centripetal term, and a sideways cross term whenever the arm stretches while it turns.

Set how the radius breathes in and out and how the angle accelerates, then read the signed vector components on a rotating arm. Finally, rebuild the motion from noisy Cartesian positions and watch angular aliasing appear.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The motion is prescribed and the position observations are simulated. The Lab does not certify forces or safety.

Physics tutorial

## A rotating basis is part of the derivative

**Background** Each motion here has three parts: the shape of the path, the timing along it and a record of observed positions.

**Why it matters** Compare the vectors drawn on the arm with the derivatives you recover from measured positions.

Start with the essentials

- **Focus question**: Where does the transverse cross term come from?
- **One-sentence intuition**: In polar coordinates you differentiate both the coefficient and the basis vector it multiplies.

### Core mathematical model

#### A radius that stays positive and a rotation of either sign

$$
\begin{aligned}r(t)&=r_0e^{dt/T}\left[1+\epsilon_r\sin\left(\frac{2\pi t}{T}\right)\right]\\\theta(t)&=\theta_0+\omega_0t+\tfrac12\alpha t^2\end{aligned}
$$

The radius stays positive because the breathing fraction is strictly less than one.

#### Differentiate the basis too

$$
\dot{\mathbf e}_r=\dot\theta\mathbf e_\theta,\qquad\dot{\mathbf e}_\theta=-\dot\theta\mathbf e_r
$$

A direction that rotates is changing, even when the coefficient in front of it stays constant.

#### Velocity components

$$
\mathbf v=\dot r\mathbf e_r+r\dot\theta\mathbf e_\theta
$$

Both the change in radius and the rotation contribute to the velocity in the inertial frame.

#### Acceleration components

$$
\begin{aligned}\mathbf a&=(\ddot r-r\dot\theta^2)\mathbf e_r\\&\quad+(r\ddot\theta+2\dot r\dot\theta)\mathbf e_\theta\end{aligned}
$$

The cross term comes purely from the kinematics. These are signed components along the basis vectors at that instant.

#### Rebuilding the motion from positions

$$
\begin{aligned}\widehat r_i&=\sqrt{x_i^2+y_i^2}\\\widehat\theta_i&=\operatorname{unwrap}(\operatorname{atan2}(y_i,x_i))\end{aligned}
$$

Fit local quadratics in time to the measured radius and the unwrapped angle. Unwrapping always picks the nearest branch, so it cannot detect a full turn that was missed.

### Common difficulties

#### A smooth curve is not the true acceleration

**Typical misconception** A smooth reconstruction or a dense plot proves you have the true acceleration.

**Better mental model** Sampling can miss turns, local fits add bias, and the largest sampled value is not a guaranteed bound on the continuous curve.

### Run the experiment

1. 01

   #### Start with a circle

   Select Uniform circle and scrub the clock.

   **What to observe:** The radial acceleration points inward, and the velocity points along the transverse direction.

2. 02

   #### Isolate the cross term

   Compare Radial only, Inward spiral and Breathing spiral.

   **What to observe:** The direction of the cross term depends on the radial motion and the sign of the rotation.

3. 03

   #### Reverse the rotation

   Select Reverse the rotation; inspect the midpoint with zero angular velocity.

   **What to observe:** At the moment the rotation reverses, the angular acceleration can still be nonzero.

4. 04

   #### Miss a turn

   Acquire Angular aliasing, then increase frame rate and acquire again.

   **What to observe:** The fit to the raw angle can look smooth and still be wrong. The analytic reference shows the motion that was missed.

Physical conclusion

Different coordinate components describe the same motion in an inertial frame. Singular frames and sparse observations each have clear limits.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)