homework and exercises - Rocket projectile motion problem

I'm not really sure where to start with this question:

In a brisk wind from the south-east of $14.1\:\mathrm{m/s}$ a rocket pointed due north is launched from a cliff that is $500\:\mathrm{m}$ above the ground. The $1000\:\mathrm{kg}$ rocket is launched at an angle at $30º$ from the horizon. On launch the rocket's engine provides a thrust that decreases with time according to the following relationship:

Thrust ($\mathrm{N}) = 10000 (5-t)$, where t is in seconds.

You may ignore air resistance. Assume that the rocket's mass remains constant while the rocket engine is burning and that $g = 10\:\mathrm{m/s^2}$. Determine (i) where the rocket will land downrange, (ii) the apex of the rocket's flight, (iii) the position of the rocket after it is in flight for $10\:\mathrm{s}$ and (iv) the direction and magnitude of the rocket's velocity after it is in flight for $10\:\mathrm{s}$. State your answers using the appropriate displacement and velocity vectors in $\vec{i}$, $\vec{j}$, $\vec{k}$.

I started by trying to draw some diagrams in the different directions, but I couldn't visualize it. Since it mentions the three vectors, does that mean I have to split it into three system of equations, then solve the unknowns that way?

Answer

As this is clearly a homework question I won't provide you with a full solution but because it's a fairly complicated problem I'll try and point you in the right direction.

Set up a reference frame of $x,y,z$ axis with origin at the point of launch, as in the diagram above.

The velocity vector $\vec{v}$ needs to be decomposed into three vectors $\vec{v_x}$, $\vec{v_y}$ and $\vec{v_z}$, which exist independently from each other.

Knowing these components allows to calculate the position vectors $\vec{x}$, $\vec{y}$ and $\vec{z}$, in time $t$.

There are two complications.

1) Wind: "You may ignore air resistance" to my mind means that the $x$ and $y$ components of $\vec{v_w}$ simply have to be added to $\vec{v_x}$ and $\vec{v_y}$, respectively. Wind thus causes the rocket to 'drift' away from the $x$ and $y$ axis.

2) Rocket burn time: your rocket motor only thrusts for $5\:\mathrm{s}$, so after $5\:\mathrm{s}$ the equations of motion change. You must therefore determine $\vec{x}$, $\vec{y}$ and $\vec{z}$ at $t=5\:\mathrm{s}$, then apply the new (no thrust) equations of motion to determine the final landing coordinates of the rocket.