Best launch angle and velocity: HANDS DOWN..

[Ether]

Originally Posted by matthewdenny

Here is my Spread Sheet. See the sheets for 'Flight Calc' and 'Flight Path' to get all the details. (You can check out the other sheets too if the interest you)

Nice job on the spreadsheet. Thanks for posting it.

Originally Posted by matthewdenny

I used .01sec intervals starting from the launch position, Calculate Vx, Vy, and Drag Force, I use Drag force and gravity to adjust the numbers for the new angle of motion and Velocity, and repeat to trace the arc.

May I offer a suggestion? The way you implemented the integration appears to have accuracy/stability issues, which could easily be fixed.

All the plots below are based on the parameters in your spreadsheet (launch speed = 35 ft/sec, launch angle = 35 deg,
g = 32.2 ft/sec², terminal velocity = 36 ft/sec, ball weight =3 lbf etc):

Plot1 shows your results compared to the true* flight path.

Plot2 shows your model with the time step changed from 0.01 to 0.005. Notice the improvement.

Plot3 shows your model with the time step changed to 0.003. It is beginning to diverge.

Plot4 shows your model with the time step changed to 0.002. It is very much off track.

Plot5 shows what is possible with a carefully-implemented Euler method at 0.01 timestep

Plot6 shows Euler with a 0.002 timestep.

Plot7 shows Trapezoidal integration of X and Y at 0.01 timestep. Notice you can't even see the red.

Trapezoidal integration is easy to do in Excel for problems of this type. I can write a short paper explaining how to do it if anyone is interested.


*mathematically "true" based on the model used. I computed the true flight path using second-order Runge-Kutta at 0.01 time step. With RK2, smaller time steps are not necessary for this application as they yield virtually identical results: See PlotRK