Catapult math

[S1nus]

Anyone have any tips on finding the angle to use on a pneumatic catapult for maximum height and distance?

[billbo911]

Sorry, I won't quote any equations except to say, 45 degrees is going to give maximum distance.

Now, maximum distance isn't always the optimal trajectory.

Putting a 24" dia. ball through a 37" opening requires fairly accurate positioning if you are throwing on a parabolic curve that is provided by a 45 deg. launch angle.

If you can't reach the bottom of the goal at 45 deg, then moving closer and increasing your launch angle will make it even harder to get the ball to pass through the goal, because the ball will be at the right height for a shorter distance.

A flatter trajectory might be a better choice as the ball will remain in the correct height range for a longer distance. Thus, you need to lower your launch angle. This also requires you to shoot a further distance.

Honestly, testing your shooter will help you figure out what exactly is your "optimal" angle.

[blaze8902]

My Team has a couple programs that allow for trajectory mapping of the ball including air resistance. It's able to be utilized to calculate the range of scoring starting distances. PM me some info on your shooter prototype and I can give you some numbers, and also the code if you want it.

I'll need the starting height of the ball, the initial velocity, and the launch angle. The program itself compares the Starting distance "x" and shows which range of those values will result in a trajectory that intercepts the goal range, a range of "y" values.

If you give us the information necessary for determining those values for your current prototype, we can help you determine what modifications to either your initial velocity or your launch angle in order to optimize or improve performance.

Many of the necessary equations can be found on this page, btw.
http://en.wikipedia.org/wiki/Trajectory_of_a_projectile

Just an example or two:
at a starting height of about two feet, the optimal angle at 10m/s launch velocity is 40* for the largest scoring range.

At 13m/s it's closer to 35* for largest range of scoring goals at the same starting height.

PM if you need help determining initial velocity or launch angle or any of that jazz.

[Ether]

My Team has a couple programs that allow for trajectory mapping of the ball including air resistance... Many of the necessary equations can be found on this page http://en.wikipedia.org/wiki/Trajectory_of_a_projectile

Were you guys able to get the your computed values to agree fairly closely with your test data using the linear model discussed in the wikipedia article?

[Ether]

Has anyone here on CD actually done a test to determine the terminal velocity of this year's game piece?

e.g. Drop the ball in still air from a sufficient height and videorecord it.

This would allow determination of the drag vs speed function.

(At least for the given air temperature and pressure).

[pfreivald]

Has anyone here on CD actually done a test to determine the terminal velocity of this year's game piece?

e.g. Drop the ball in still air from a sufficient height and videorecord it.

This would allow determination of the drag vs speed function.

(At least for the given air temperature and pressure).




We didn't. We just made a guesstimate on reasonable values for our initial calculations, and used those as a starting point for our prototyping. Once we got into prototyping, experiment ruled over theory.

[Ether]

Once we got into prototyping, experiment ruled over theory.

It's not about "theory" really. It's about how to build a good simulation. If you have adequate empirical data to establish real-world parameter values for a simulation model, the model can be very useful.

[pfreivald]

It's not about "theory" really. It's about how to build a good simulation. If you have adequate empirical data to establish real-world parameter values for a simulation model, the model can be very useful.




True enough!

[Michael Hill]

Here's one of the fun oddities of this year's game.

Let's assume the ball is a sphere (it's close enough for this purpose)

Drag on a sphere is based on the Reynolds Number as seen here: http://www.grc.nasa.gov/WWW/k-12/airplane/dragsphere.html

To calculate the Reynolds Number, the formula is:

Re = ρvc/μ

Where:
Re = Reynolds Number
ρ = Fluid Density
v = Velocity
c = Characteristic Length
μ = Dynamic Viscosity

For air (We're assuming 20deg C):
ρ = 1.2041 kg/m^3
μ = 1.983*10^-5 kg/(m-s)

For the ball:
c = 25 inches = 0.635 m

So our formula for Re is now

Re = (1.2041 kg/m^3)*v*(0.635 m)/(1.983*10^-5 kg/(m-s))

Re = 3.8558 * 10^4 * v

Where v = velocity in m/s

I would say our ball is a fairly rough ball given the wrinkles and the material, so look more toward the rough line on the chart.

From v = 1 m/s to 5 m/s, we have Reynolds Number values between 3.8558 * 10^4 and 1.9279 * 10^5.

We pass straight through the transition between laminar and turbulent flow. So what does this mean? For some of the "hard" shooters, they're going to experience a transition from turbulent to laminar flow as the ball slows down causing potentially unpredictable results.

That's one of the reasons I like this game.

[Ether]

Here's one of the fun oddities of this year's game...

Thanks for posting this calculation Michael. The NASA page you linked was quite interesting too.

[blaze8902]

Were you guys able to get the your computed values to agree fairly closely with your test data using the linear model discussed in the wikipedia article?




Sorry for reply delay.
Yeah, we were. Although we found that while doing angle optimization, the real challenge became in making sure that the level of control of the release angle was high enough. We found that it became necessary to construct additions to the catapult arm that we were using, and that the ending angle of the catapult arm didn't usually result in the right release angle. (The ball was rolling off the arm before the arm finished it's motion)

[Ether]

What value of "k" did you determine for this year's game piece to give the best match between your test data and the model?