# Golf Balls and differential equations

1. Oct 12, 2012

### Hercuflea

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):

1. The problem statement, all variables and given/known data
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$_{x}$ = $\frac{dv_{x}}{dt}$
and
-.25v$_{y}$ -g = $\frac{dv_{y}}{dt}$

Therefore

$\frac{dv_{y}}{dt}$ +.25 v$_{y}$ = -g where g is the earth's acceleration due to gravity.
and
$\frac{dv_{x}}{dt}$ +.25v$_{x}$ = 0

2. Relevant equations

Integrating factor: e$^{\int P(t) dt}$
x range = v$_{i}$cosθ * t

3. The attempt at a solution

For v$_{x}$:

I(t) = e$^{\int P(t) dt}$
I(t) = e$^{.25t + k_{1}}$

$\int(d e^{.25t}e^{k_{1}}v_{x} /dt)$ = $\int 0 dt$

e$^{.25t}$e$^{k_{1}}$v$_{x}$ = C$_{1}$

v$_{x}$ = C$_{1}$e$^{.25t}$ because e$^{k_{1}}$ is just a constant too.

v$_{i}$cos($\Theta$) = C$_{1}$e$^{.25t}$

I use the statutory initial velocity of a golf ball of 76.2 m/s.

cos($\Theta$) = $\frac{C_{1}}{76.2}$e$^{.25t}$

$\Theta$ = cos$^{-1}$($\frac{C_{1}}{76.2}$e$^{.25t}$)

For v$_{y}$: (skipping the prelim stuff)

e$^{.25t}$e$^{k_{2}}$v$_{y}$ = -gt + C$_{2}$

v$_{y}$ = e$^{-.25t}$e$^{-k_{2}}$(-gt + C$_{2}$)

v$_{i}$sin(θ) = e$^{-.25t}$e$^{-k_{2}}$(-gt + C$_{2}$)

sin(θ) = $\frac{e^{-.25t}e^{-k_{2}}(-gt + C_{2})}{76.2}$

θ = sin$^{-1}$($\frac{e^{-.25t}e^{-k_{2}}(-gt + C_{2})}{76.2}$)

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$_{y}$ differential equation? Laplace Transform?

2. Oct 12, 2012

### Hercuflea

I tried to solve v$_{y}$ using Laplace Transforms.

I got

v$_{y}$ = e$^{-.25t}$(v$_{y}$(0) +4g) - 4g

I still dont know v$_{y}$(0) because the angle can vary.

3. Oct 13, 2012

### Hercuflea

Getting closer, can someone tell me if there is a way to write initial velocity in y as a function of θ?