Putting a golfball on a plane, but not directly up or down the plane

[Labeno]

I’m trying to figure out what may be very simple but I can’t find any examples online or in Physics books. The goal is to determine the final location of a ball (golfball) rolling on an inclined plane (putting green). I know a putting green is very curved, but I’m trying to solve a simple case assuming the green is a plane tilted at an angle. The golfball is putt in an arbitrary direction not directly up or down the planes incline. I attached a picture of what I mean. In this case the ball is hit up the incline but to the right of straight up the plane.

The putting green surface has enough friction to eventually stop the ball, so it’s not a frictionless problem, but the friction is constant at all speeds of the ball. Also, assume the putting green is not so steep that friction is not enough to slow the ball down.

I’ve been studying free body diagrams fine, and if the ball goes straight up or down the incline, then the free body diagrams describes it perfectly and I can derive the final location of the ball knowing the initial velocity of the putt.

So by not hitting the ball directly up or down the ramp, it adds a dimension to the calculation. Does this mean it becomes a projectile calculation or something else?

I’m trying to keep this as simple as possible. The known values of this problem are:
- Initial velocity (speed and direction): Vo
Coefficient of rolling friction of the ball on the green: mu = .116
Mass of the ball: .0459 kg
Gravity: 9.8 meters per second squared
Golfball is just a point (keeping it simple)
Golfball is rolling and not sliding (assume rolling from start of hit to keep it simple).
Surface of green is not too steep, i.e. friction will always slow it down.

Goals:
Determine the final location of the ball
Determine time to get from start to final location
Determine where the ball is at any point in time (need parametric equations for this).

I’m at a loss, so any help is appreciated.

 

[K^2]

If you ignore angular momentum of the golf ball, this is just a problem of projectile motion. However, projectile motion with drag/friction is a complicated problem. The only type of drag that's easy to solve with is linear drag. (Drag proportional to velocity.) Constant friction results in coupling between your equations of motion along the incline and perpendicular to it. So you end up with a system of non-linear equations, and these aren't fun.

 

[Labeno]

Thanks for the helpful response.
I found the following web link showing the parametric equations for projectile motion with linear drag.

http://www.damtp.cam.ac.uk/user/stcs/courses/dynamics/handouts/handout3.pdf

I guess this would be the non linear equations you are referring too? I agree, they aren't fun. Anyways, since the ball is not actually launched in the air, then gravity is not working at full force. When the plane incline is close to zero, gravity will have almost no effect. Would the new gravity = g * sin(angle of ramp)?
Also, from the above web link, does 'k' refer to the coefficient of friction (in this case .116), or is it something else?

Thanks again for helping.

 

[K^2]

Oh, no. These are linear equations. These are the nice ones. The ones you get with constant friction force are much worse.

With linear drag, the force varies with velocity, so the ball will never actually come to a complete stop. It will eventually just roll downhill at some constant speed. That's not really what you want. Just something that can be solved comparatively easily.

To account for gravity, yes. You'll act as if g = actual g * sin(angle). That's along the incline. Perpendicular to it, the ball is affected by friction only. That sets up a system of differential equations that are a pain to solve. But I do believe there is a general solution that includes this case. It's just not a nice one.

 

[Labeno]

Bummer. You'd think this type of problem was much easier considering projectile motion without friction is so easy, and free body diagrams with friction are so easy in one dimension. If you happen to find a website that describes this situation, I'll be grateful if you post it here.
Thanks for you help.

 

[Chestermiller]

With all due respect to K^2, I think that this problem can be solved numerically without much trouble. It will involve a set of 4 coupled non-linear first order ODEs, which can be handled using an automatic integrator. I feel that the real complexity in this problem will be in accurately parameterizing the drag from the putting green grass.

 

[Labeno]

Thanks for the interest. Do you know of a website or tutorial that can guide me on this specific problem? Thanks.

 

[Chestermiller]

Thanks for the interest. Do you know of a website or tutorial that can guide me on this specific problem? Thanks.

If you have the knowhow to formulate the ODEs using Newton's second law (including rotational inertia of the ball), you will need a tool for numerically integrating the equations. There are free numerical integrators available over the internet (with time step and error control), and some of the math packages like mathematica may have numerical integrators. Talk to some of your contacts about numerical integrators that they like.

 

[K^2]

Oh, if you are happy with numerical solution, certainly. It's very easy to do. It's only the analytical solution that's difficult in this case.

For numerical solution, if you have access to Mathematica, it's very straight forward. NDSolve method will straight up solve this if you give it the right equations and starting conditions. If you know a little bit about programming, you can even write your own code. Runge-Kutta (The RK4) should be more than sufficient here, so long as you make the time increment small enough.

 

[Labeno]

I feel little overwhelmed with trying to learn an ODE or Mathematica. I'm a c coder trying to write a simple putting game for the iPhone, and thought there would be a simple equation for this since the one dimensional equation is trivial. Add a little lateral motion did not occur to me to be so non-trivial. Is there a simple equation out there that does a good job estimating the projectile motion versus the non trivial but exact equation?
Thanks for the great help.

 

[Chestermiller]

I feel little overwhelmed with trying to learn an ODE or Mathematica. I'm a c coder trying to write a simple putting game for the iPhone, and thought there would be a simple equation for this since the one dimensional equation is trivial. Add a little lateral motion did not occur to me to be so non-trivial. Is there a simple equation out there that does a good job estimating the projectile motion versus the non trivial but exact equation?
Thanks for the great help.

Try Forward Euler numerical integration with very small time steps.

 

[K^2]

You really should have started with it being for a game.

Forward Euler tends to have very bad results in physics. Though, in this case, you have no collisions, so maybe it's not too bad.

What works much, much better for simple simulations like that is Velocity Verlet. It's extremely simple if you ignore all the "why it works" math stuff. The equations there aren't terribly useful as they are written. Instead you compute the velocity update at the beginning of next time step. So it looks like this.

\vec{v}(t) = \vec{v}(t-\Delta t) + \frac{\vec{a}(t-\Delta t) + \vec{a}(t)}{2}\Delta t
\vec{x}(t+\Delta t) = \vec{x}(t) + \vec{v}(t) \Delta t + \frac{1}{2}\vec{a}(t)\Delta t^2

This doesn't correct for a(t) being function of v(t), but that's a very minor error with sufficiently small \Delta t.

In practice, the way you implement this is as follows. For every time step: (Pseudo-code, assume vector operations.)

\\Compute acceleration vector a using direction of motion, friction coefficient, local slope, and gravity.

v+=0.5*(a + a_old)*dt;
x+=v*dt + 0.5*a*dt*dt;
a_old=a;

Rinse, repeat.

It's an extremely simple, yet rather powerful algorithm for integration of equations of motion. The reason it's used in just about every physics-based game out there is because the errors this introduces are pretty much random, and tend to cancel out, giving you nice trajectories.

 

[Labeno]

Thanks a ton.
I will spend time studying this and doing various tests over the next few days.