Euler's Method for motion of golf ball with lift

[Weskhan]

Homework Statement

We were given two equations of motion for a golf ball with lift. We must figure out the maximum range using an initial velocity of 255 ft/sec. I'll just retype what's on the sheet.

"Lets incorporate lift. The free body diagram with all the forces shown is depicted in figure 3. Recall lift is a linear function of velocity, CL = k/V, k is a constant. The ration of k/m, where m is the mass of the golf ball, has been found experimentally to be 0.247 sec^-1. The equations of motion are now

1. -cVx - kVsin(angle) = mdVx / dt = -0.25Vx - 0.247Vy = dVx / dt
2. -cVy + kVcos(angle) - mg = mdVy / dt = -0.25Vy + 0.247Vx - g = dVy/dt

These are now coupled equations where Vy depends on Vx and Vx depends on Vy. Sorry there is no simple analytic solution to these equations. So we will solve them numerically. HOwever, this is crucial, we do have an analytic solution for k=0. Thus we can check our numerical solution with the k=0 case and see if we recover the analytic solution. This is a standard technique in science and engineering, always check a numerical solution against an analytic solution. Then you can have confidence in your numerical solution. The simplest way to solve equations like the ones we have is the Euler method. Basically you break the solution into small steps, where the solution at the nth step depends on the solution at the nth-1 step. You should be able to implement on Euler solution on any PC. You can adjust the time step and see how the results depend on the choice of chance in t. Compute numerical solutions for angles starting at 4 degrees to 16 degrees, say every 2 degrees and determine the maximum range using an initial velocity of 255 ft/sec. Once you know how to determine the maximum range, find the difference in range between Atlanta and Laramie. You will have to track down the difference in air density between the two altitudes."

Homework Equations

1. -cVx - kVsin(angle) = mdVx / dt = -0.25Vx - 0.247Vy = dVx / dt
2. -cVy + kVcos(angle) - mg = mdVy / dt = -0.25Vy + 0.247Vx - g = dVy/dt

The Attempt at a Solution

All we have got is the Euler equation for this because he helped us with it.

dx / dt = V
xi+1 - xi / ti+1 - ti = Vi
-0.25Vx - 0.247Vy = dVx/dt

Vxi = t1 - t0 [f1(Vxi, Vyi) + Vxi)] *So I think this is the Euler's equation for this. I kind of understand how to do Eulers, but not on equations like this. If anybody can help please do. Thank you.

 

[BigJDubs]

-0.25Vyi + 0.247Vxi - g = dVy/dtVyi = t1 - t0 [f2(Vxi, Vyi) + Vyi)] *Again, I think this is the Euler's equation for this, but help is appreciated. Thank you.

 

[nlightner]

Thank you for sharing your homework problem and attempts at a solution. It seems like you are on the right track with using Euler's method to solve the coupled equations of motion for the golf ball with lift. Euler's method is a numerical approach to solving differential equations, where you break the solution into small steps and use the previous step's solution to calculate the next step's solution. This can be done on a computer or by hand, but it can be more efficient and accurate to use a computer program.

In this case, you have two equations of motion that are coupled, meaning that the velocity in one equation depends on the velocity in the other equation. This makes it more challenging to solve analytically, which is why you are using a numerical approach. It is important to note that Euler's method is an approximation and may not give an exact solution, but it can give a good estimate.

To apply Euler's method to this problem, you will need to choose a time step, which is the amount of time that passes between each calculation. This can be adjusted to see how the results change with different time steps. You will also need to use the initial conditions given in the problem, such as the initial velocity and angles, to calculate the solution at each time step.

Once you have the numerical solutions for the different angles, you can determine the maximum range by finding the point where the ball reaches its furthest distance. This can be done by plotting the distance traveled over time and finding the maximum point, or by using a computer program to find the maximum value.

Lastly, you will need to consider the difference in air density between Atlanta and Laramie to determine the difference in range. This can be calculated using the given information and equations for air density at different altitudes.

In summary, Euler's method is a useful numerical approach to solving coupled equations of motion, and it is important to check the numerical solution against an analytic solution for validation. I hope this helps in your understanding of the problem. Good luck with your homework!