Motion of a particle - Your turn

The motion of a particle in three dimensions is given by the following parametric equations:

\[\begin{split} \begin{align*} x(t) &= \cos(t) \\ y(t) &= 2\sin(t) \qquad \text{for} \quad 0 \le t \le 2\pi \\ z(t) &= t \end{align*} \end{split}\]

Part (a)

Redo all five steps illustrated in the example.

# TODO: Write your solution below

Part (b)

Verify that the magnitude of your tangent vector at \(t = 0\) is unity.

# TODO: Write your solution below

Part (c)

Plot the speed, acceleration, and the angle between the velocity and the acceleration vectors as a function of time using the plt.subplots() command. Comment on the values obtained for the acceleration for \(t\) equal to \(0\) and \(2\pi\).

Hint: You’ll want to add some plot labels and include layout='constrained' for proper spacing!

# TODO: Write your solution below

Exporting your work

When you’re ready, the easiest way to export the notebook is to File > Print it and save it as a PDF. Remove any excessively long, unrelated outputs first by clicking the arrow → next to the output box and then Show/hide output. Obviously don’t obscure any necessary output or graphs!