Projectile Motion from a height with Air Resistance

[RawrSpoon]

Hey all, me again. This time my question has to do with projectile motion with air resistance from a given height.

Homework Statement

A cannon is located on a cliff of height h. If the muzzle velocity of a projectile is v₀, find the range of the projectile when the drag is proportional to the velocity.

Homework Equations

ma=-mg-kv_y. Vertical motion
ma=-kv_x. Horizontal motion
mg=kv_t. Terminal velocity

The Attempt at a Solution

I first attempted to find t via the horizontal motion formula, but my answer doesn't take into account the height difference. Then I tried integrating the vertical motion equation and got an answer of
t=-v_t/g * ln((v_t+v_y)/(v_t+v_oy)) where v_oy is the initial vertical velocity. I then made v_y=dy/dt and tried to find t from there but I have no idea how to do the integration.

Am I even on the right path? I'm hoping if I can find the time t given y I can plug that into xcosθ*t to give me the range.

 

[rude man]

ƩF_x = md²x/dt² → first 2nd-order linear diff. eq. in x
ƩF_y = md²y/dt² → second linear diff. eq. in y

You know the initial conditions (4 of them for the two equations) & they're linear constant-coefficient, so solution is straight-forward.

(I would use Laplace transform but that's 'cause I'm an EE. EE's can't so much as tie their shoelaces
without transforming to Laplace! )