Task 3

The Descent protocol

Points for this task: 25

The Descent Protocol

Premise

You’re a companion of Han Solo, the Millennium Falcon took a hit during a short battle near the ruins of a forgotten Sith outpost orbiting a rogue planet. While escaping, you made the dumb call to jump into an uncharted gravity rift. Now you’re trapped in what appears to be a vertical gravity tunnel, the data does not match anything you seen before. Gravity doesn’t pull in one direction. It changes with time. Your sensors spit out a equation:

\[g(y) = 9.8 \cos(2t)\]

The ship’s boosters are offline, and Chewie’s unconscious. Han is piloting manually, he needs your help in finding the trajectory.
Suddenly, your mind is touched by a vision from Master Yoda.
The vision says:

\[\begin{aligned} k_1 &= f(x, t) \\ k_2 &= f\left(x + \frac{h}{2} k_1, \; t + \frac{h}{2}\right) \\ k_3 &= f\left(x + \frac{h}{2} k_2, \; t + \frac{h}{2}\right) \\ k_4 &= f\left(x + h \cdot k_3, \; t + h\right) \\ x_{\text{next}} &= x + \frac{h}{6} \left(k_1 + 2k_2 + 2k_3 + k_4\right) \end{aligned}\]

System to save:

\[x(t) = \begin{bmatrix} y(t) \\ v(t) \end{bmatrix}\]

With dynamics as:

\[\frac{d}{dt}x = f(x, t) = \begin{bmatrix} v(t) \\ -9.8 \cos(2t) \end{bmatrix}\]

You realize you are 10 V (Vectras) above the abyss, the ship is still and each pulse of time is 0.1 seconds.

Use Yoda’s vision to calculate the state ( x ) at ( t = {0.1} ) seconds and save yourself.

Refer here to understand Yoda’s vision

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