Projectile motion. Relationship between R and theta

[kevinfoss7]

As a part of a lab, we were asked to find the theoretical relationship between R (the range of the projectile) and theta (the launch-angle). There is no recorded value for time.


equations given are:

R(θ)=x=d/dt v0x = v0xt-->R(θ)=v0xt v0x=v0cosθ

z= d/dt(v0z-gt)+z0=v0zt-1/2(gt^2)+z0 v0z=v0sinθ

anything with a 0x or 0z after means initial in that direction. I am unsure how to make those subscript on the forum


I have the equation simplified to

R(θ)=(v0cosθ)*(- v0sinθ±((v0sinθ)^2+2g(z-z0))^1/2))/g

but the presence of z-z0 is bothersome... how can I reasonably cancel it out?

 

[kevinfoss7]

sorry. Simpler way to state the given equations

R(θ)=v0cosθt

z=v0sinθt-1/2(gt^2)+z0

 

[kevinfoss7]

solved for t

t=(-v0sinθ±( (v0sinθ)^2+2gz)^1/2)/g

t=R(θ)/(v0cosθ)