Tool · Sandbox
Motion Explorer
Choose a motion, or type your own \(x(t)\) and \(y(t)\), and watch its velocity and acceleration as it moves. Switch between rectangular and path components, show the moving \(\et, \en\) frame and the osculating circle, and read every quantity live.
Choose a motion
How to use the Explorer
Choose a motion, then press Play, drag the time slider, or drag the particle along its path. Reset goes back to the start, and Playback runs the motion at half or double speed. The page address (the part after #) keeps the motion you chose, including a custom one, so a bookmark or a copied link brings it back.
Components
Components splits the vectors at the particle, drawn tip to tail. Rectangular shows \(v_x, v_y\) and \(a_x, a_y\) along the fixed \(\ihat\) and \(\jhat\) (Lesson 3). Path shows \(a_t\) along \(\et\) and \(a_n\) along \(\en\) (Lesson 6); the velocity needs no split, because \(\vvec = v\,\et\). Under Show, turn \(\vvec\), \(\avec\), the moving frame \(\et, \en\), the osculating circle and the trace on and off.
Live values
The State, Velocity and Acceleration cards list the position, the direction of \(\vvec\), the radius of curvature \(\rho\) and every component of \(\vvec\) and \(\avec\), to 4 significant figures. To check a hand calculation, pause at the same \(t\) and compare.
Typing your own motion
Choose Custom and type \(x(t)\) and \(y(t)\) in metres, using t for time in seconds. The Explorer differentiates numerically, so any smooth function works.
| Motion | \(x(t)\) | \(y(t)\) |
|---|---|---|
| Projectile, 20 m/s at 50° | 20cos(50°)t | 20sin(50°)t - 4.905t^2 |
| Circle of radius 3, ω = 1 rad/s | 3cos(t) | 3sin(t) |
| Worksheet Problem 1 | 4t - t^2 | 3t^2 |
| Slider on y = 0.1x², vx = 2 | 2t - 6 | 0.1(2t - 6)^2 |
| Spiral | (0.5 + 0.3t)cos(t) | (0.5 + 0.3t)sin(t) |
Functions: sin cos tan sqrt exp ln abs; constants pi and e; powers with ^. Multiplication can be implied (2t, 3cos(t)).
Ideas to try
Each idea loads a motion into the Explorer and sets what it shows.
Where \(a_t\) changes sign
A projectile slows on the way up and speeds up on the way down, while \(\avec\) stays \(9.81\ \text{m/s}^2\) straight down. With path components on, where is \(a_t\) negative, zero and positive? Where is \(\rho\) smallest?
A circle at constant speed
Radius \(3\ \text{m}\) at \(3\ \text{m/s}\). Confirm that \(a_t = 0\) and \(a_n = v^2/R = 3\ \text{m/s}^2\) all the way round, always toward the center.
Speeding up on a circle
The same circle, starting at \(1\ \text{m/s}\) with \(a_t = 1\ \text{m/s}^2\). \(a_t\) stays fixed while \(a_n = v^2/R\) grows, so \(\avec\) swings toward the center.
The osculating circle of an ellipse
\(x = 4\cos t,\ y = 2\sin t\), with the osculating circle on. Where is the circle smallest? Is \(\avec\) ever perpendicular to \(\vvec\)?
Stopped, but still accelerating
A point on the rim of a rolling wheel stops for an instant each time it touches the ground. Step to that instant: \(\vvec = \mathbf{0}\), yet \(\avec\) is not zero. Which way does it point?
Two independent motors
A pen plotter draws a figure eight, \(x = 4\sin t,\ y = 2\sin 2t\). In rectangular components each motor's part is a simple oscillation of its own.
Check Worksheet Problem 1
The drone of Worksheet Problem 1, \(x = 4t - t^2\), \(y = 3t^2\). It opens paused at \(t = 1\ \text{s}\): compare the speed and \(|\avec|\) with your answers to part (b).
Your own motion
Enter a problem from the Practice Lab as a custom motion, then check your answers against the live values.