Determine path of golf ball
All the below is based on the paper the physics of putting from <PERSON>. The goal is to determine the path of a golf ball.
A golf ball is launched with velocity and a launch angle of towards a hole along the y-axis. The "green" is level.
We know that the components of are
$$ v_x = v \cdot \sin{(\beta)}$$
and
$$ v_y = v \cdot \cos{(\beta)}$$
If $\beta$ is 0 then $v_x$ is also equal to 0.
In this case, the golf ball stops when $v_y$ is 0. 1002
In order to determine when $v_y$ is equal to 0, we need to calculate $a_y$. This is pretty straightforward if we know the coefficient of rolling friction, $\rho$.
However, I want to determine the path for a "green" where slopes are present.The surface of the green is sloped at angles, with respect to the x-axis, $\theta$ , and $\phi$ along the y-axis.
So, if it is a straight uphill put, $\theta$ will be equal to 0 and $\phi$ will be larger than 0.
According to the mentioned paper we have the following formulas for acceleration:
$$ a_x = -1 \cdot g \cdot sin{(\theta)} - (f \cdot sin{(\phi)}) / m$$
and
$$ a_y = -1 \cdot g \cdot cos{(\theta)} \cdot sin{(\phi)} - (f \cdot cos{(\phi)}) / m$$
In addition, the direction and magnitude of the tangential component of the contact force are given by:
$$\tan(\alpha) = \frac{\rho\cdot\cos(\theta) \cdot\cos(\phi)\cdot\sin(\beta)-I\cdot\sin(\theta)}{\rho \cdot\cos(\theta) \cdot\cos(\phi) \cdot\cos(\beta)- I\cdot\cos(\theta) \cdot\sin(\phi)} $$
and
$$f = \frac{\rho\cdot\cos(\theta) \cdot\cos(\phi)\cdot\cos(\beta)-I\cdot\cos(\theta)\cdot\sin(\phi)}{(1 + I)\cdot\cos(\alpha)} \cdot mg$$
It is assumed that
$$ \rho = 0.131 $$
and the moment of intertia of the ball is
$$ I = 0.40 $$
According to the author
"The above expressions for f and φ along with $a_x$ and $a_y$ will allow the x- and y-components of the acceleration of the golf ball to be determined for greens of various slopes. Given these accelerations, along with the initial launch conditions, the paths of the putted golf balls can be determined."
My question is when does the ball stop? With the level green it is easy; the ball stops when $v_x$ and $v_y$ are both equal to zero. However, if we have straight uphill green where $\phi$ is large the ball will stop at some point on the hill, where $v_y$ is 0, but the ball will start roll back towards the starting location because of the steepness of the hill.
What is sort of speak the "end criteria"? 1002