Is this the end?

This is also gonna be a brief scroll, mainly only results and discussion.

This part will perhaps be the part that is easiest and closest to what you already know from Physics 1 and Physics 2 so it will be even shorter.

We have a model for air resistance given by:

\(F_d = \frac{1}{2}\rho C_d A v_{\text{drag}}^2\)

The Force is the air resistance experienced by a body with A as cross sectional area. \(v_{\text{drag}}\) is the velocity of the body with respect to the atmosphere. Rho is the density of the atmosphere at that specific point, and C the drag coefficient.

This is the heart and soul of this scroll as it will be used at almost every step.

As we know, the angular velocity is just change in theta in change of time

\(w = \Delta \theta \over \Delta t\)

Given that the entire atmosphere has the same angular velocity. 

v drage is simply the velocity minus the rotation of the planet, this is due to the fact that our planet rotates anticlockwise, otherwise it would've been a + there instead of a minus. 

After a while after entering the atmosphere we will reach equilibrium in the radial direction, but we will keep braking tangentially as long as the velocity of the air is different from us, until at some point we will have the same velocity as the air, so after a while our velocity in the tangential component will be stable. As mentioned our radial velocity will simply be at equilibrium because the forces will after a while sum up to zero and we will reach terminal velocity. 

If we assume tangential velocity is equals to 0, we can see these equations that use the fact that we know terminal velocity means the sum of forces are equal. 

Now onto simulating the the descent, we have to fin the acceleration and we did the following to find it 

Our method to descend the star is simply to rotate the vector by a bit and so on. 

Here is the trajectories:

Well I'm not sure what to discuss with you here to be honest, we just landed successfully, we don't need to check if the results are realistic or anything, we made it. Maybe you would like too see our velocity graph? Here it is 

Oops the axes were shifted. We're chilling. Thanks and see you next time.