# Projectile Motion from a height with Air Resistance

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-kvy. Vertical motion
ma=-kvx. Horizontal motion
mg=kvt. 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=-vt/g * ln((vt+vy)/(vt+voy)) where voy is the initial vertical velocity. I then made vy=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.

Related Advanced Physics Homework Help News on Phys.org
rude man
Homework Helper
Gold Member
ƩFx = md2x/dt2 → first 2nd-order linear diff. eq. in x
ƩFy = 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! )