- #1
Hercuflea
- 596
- 49
Hey Everybody, I am supposed to model the trajectory of a golf ball. I have been given the equations for velocity as a function of its derivative with respect to time. I am supposed to find the x-range as a function of the angle θ. (Pardon my bad latex skills, I will fix mistakes):
These are the equations which have a mathematical solution, and they do not include lift. -.25 is the drag coefficient on the golf ball.
-.25v[itex]_{x}[/itex] = [itex]\frac{dv_{x}}{dt}[/itex]
and
-.25v[itex]_{y}[/itex] -g = [itex]\frac{dv_{y}}{dt}[/itex]
Therefore
[itex]\frac{dv_{y}}{dt}[/itex] +.25 v[itex]_{y}[/itex] = -g where g is the Earth's acceleration due to gravity.
and
[itex]\frac{dv_{x}}{dt}[/itex] +.25v[itex]_{x}[/itex] = 0
Integrating factor: e[itex]^{\int P(t) dt}[/itex]
x range = v[itex]_{i}[/itex]cosθ * t
For v[itex]_{x}[/itex]:
I(t) = e[itex]^{\int P(t) dt}[/itex]
I(t) = e[itex]^{.25t + k_{1}}[/itex]
[itex]\int(d e^{.25t}e^{k_{1}}v_{x} /dt)[/itex] = [itex]\int 0 dt[/itex]
e[itex]^{.25t}[/itex]e[itex]^{k_{1}}[/itex]v[itex]_{x}[/itex] = C[itex]_{1}[/itex]
v[itex]_{x}[/itex] = C[itex]_{1}[/itex]e[itex]^{.25t}[/itex] because e[itex]^{k_{1}}[/itex] is just a constant too.
v[itex]_{i}[/itex]cos([itex]\Theta[/itex]) = C[itex]_{1}[/itex]e[itex]^{.25t}[/itex]
I use the statutory initial velocity of a golf ball of 76.2 m/s.
cos([itex]\Theta[/itex]) = [itex]\frac{C_{1}}{76.2}[/itex]e[itex]^{.25t}[/itex]
[itex]\Theta[/itex] = cos[itex]^{-1}[/itex]([itex]\frac{C_{1}}{76.2}[/itex]e[itex]^{.25t}[/itex])
For v[itex]_{y}[/itex]: (skipping the prelim stuff)
e[itex]^{.25t}[/itex]e[itex]^{k_{2}}[/itex]v[itex]_{y}[/itex] = -gt + C[itex]_{2}[/itex]
v[itex]_{y}[/itex] = e[itex]^{-.25t}[/itex]e[itex]^{-k_{2}}[/itex](-gt + C[itex]_{2}[/itex])
v[itex]_{i}[/itex]sin(θ) = e[itex]^{-.25t}[/itex]e[itex]^{-k_{2}}[/itex](-gt + C[itex]_{2}[/itex])
sin(θ) = [itex]\frac{e^{-.25t}e^{-k_{2}}(-gt + C_{2})}{76.2}[/itex]
θ = sin[itex]^{-1}[/itex]([itex]\frac{e^{-.25t}e^{-k_{2}}(-gt + C_{2})}{76.2}[/itex])
These equations for θ seem pretty nasty, not to mention I have no way of knowing the Constants because I only know the absolute value of the velocity, not the components.
Also, these equations I have found for θ have seemingly nothing to do with range, they are a function of time. Any hints? Should I use another solution method for the v[itex]_{y}[/itex] differential equation? Laplace Transform?
Homework Statement
These are the equations which have a mathematical solution, and they do not include lift. -.25 is the drag coefficient on the golf ball.
-.25v[itex]_{x}[/itex] = [itex]\frac{dv_{x}}{dt}[/itex]
and
-.25v[itex]_{y}[/itex] -g = [itex]\frac{dv_{y}}{dt}[/itex]
Therefore
[itex]\frac{dv_{y}}{dt}[/itex] +.25 v[itex]_{y}[/itex] = -g where g is the Earth's acceleration due to gravity.
and
[itex]\frac{dv_{x}}{dt}[/itex] +.25v[itex]_{x}[/itex] = 0
Homework Equations
Integrating factor: e[itex]^{\int P(t) dt}[/itex]
x range = v[itex]_{i}[/itex]cosθ * t
The Attempt at a Solution
For v[itex]_{x}[/itex]:
I(t) = e[itex]^{\int P(t) dt}[/itex]
I(t) = e[itex]^{.25t + k_{1}}[/itex]
[itex]\int(d e^{.25t}e^{k_{1}}v_{x} /dt)[/itex] = [itex]\int 0 dt[/itex]
e[itex]^{.25t}[/itex]e[itex]^{k_{1}}[/itex]v[itex]_{x}[/itex] = C[itex]_{1}[/itex]
v[itex]_{x}[/itex] = C[itex]_{1}[/itex]e[itex]^{.25t}[/itex] because e[itex]^{k_{1}}[/itex] is just a constant too.
v[itex]_{i}[/itex]cos([itex]\Theta[/itex]) = C[itex]_{1}[/itex]e[itex]^{.25t}[/itex]
I use the statutory initial velocity of a golf ball of 76.2 m/s.
cos([itex]\Theta[/itex]) = [itex]\frac{C_{1}}{76.2}[/itex]e[itex]^{.25t}[/itex]
[itex]\Theta[/itex] = cos[itex]^{-1}[/itex]([itex]\frac{C_{1}}{76.2}[/itex]e[itex]^{.25t}[/itex])
For v[itex]_{y}[/itex]: (skipping the prelim stuff)
e[itex]^{.25t}[/itex]e[itex]^{k_{2}}[/itex]v[itex]_{y}[/itex] = -gt + C[itex]_{2}[/itex]
v[itex]_{y}[/itex] = e[itex]^{-.25t}[/itex]e[itex]^{-k_{2}}[/itex](-gt + C[itex]_{2}[/itex])
v[itex]_{i}[/itex]sin(θ) = e[itex]^{-.25t}[/itex]e[itex]^{-k_{2}}[/itex](-gt + C[itex]_{2}[/itex])
sin(θ) = [itex]\frac{e^{-.25t}e^{-k_{2}}(-gt + C_{2})}{76.2}[/itex]
θ = sin[itex]^{-1}[/itex]([itex]\frac{e^{-.25t}e^{-k_{2}}(-gt + C_{2})}{76.2}[/itex])
These equations for θ seem pretty nasty, not to mention I have no way of knowing the Constants because I only know the absolute value of the velocity, not the components.
Also, these equations I have found for θ have seemingly nothing to do with range, they are a function of time. Any hints? Should I use another solution method for the v[itex]_{y}[/itex] differential equation? Laplace Transform?