Determine flywheel velocity for ball exit velocity

I wonder if there is any formula I can use to determine how fast I need to spin the flywheel in order to make the ball exit velocity a certain speed.

It depends on your design.

A wheel working against a fixed hood will have roughly 1/2 the surface speed of the wheel imparted on the ball; the ball is rolling against the hood - relative speed of the ball contacting the hood is close to 0 while the part touching the wheel is close to wheel speed (I say “close to” because things are slipping around but for practical computations, 0 and the surface speed of the wheel are close enough).

Two opposing wheels will launch at roughly the surface speed of the wheels.

Note that the key is not RPM, it’s surface speed of the wheel (which you can figure out from the RPM and the circumference). It’s the surface of the wheel that is imparting the force so the linear speed of the surface is key. RPM is how you measure it but it’s really about surface speed of the wheel.

Things like compression and the like come into play as to how efficient all this really is so in the end, empirical testing, measurement, and tuning comes into play.

Using 1/2 the surface speed of your shooting wheel is a great start. We also did some field testing as well. Initially we were interested in the launch angle. Later we were fine tuning our launch speed. This is a frame from one of our earlier testing videos. Later we filmed in slow-mo and held a phone running a timer in frame for better time accuracy. The board is marked off in 6” increments and we measured and interpolated for position, but if you had more patience to draw all the lines you could definitely use a finer grid.


While this is a good place to start, there are unfortunately many more variables at play with a flywheel shooter. The best way to determine your projectile exit speed(muzzle velocity) is through prototyping and deriving the speed experimentally, using a slow motion camera and an object of known size. Wheel slip is often way more than people realize, and is driven by the speed of the flywheel, the coefficient of friction(CoF) between the object and the wheel, and the CoF between the object and the back of the hood. That’s why many teams used grippy materials, such as yoga mats, on the backs of their hoods. My team actually had an interesting scenario this year where we could increase muzzle velocity slightly by lowering the speed of the flywheel, because the higher speed of the flywheel was causing more wheel slip.

The Dynamics of Ball Flight and Design of a Robotic Ball Shooter | by Sajiv Shah | Towards Data Science

I don’t know if this necessarily will solve your problem, but I recommend reading through all 3 parts of this. They tried to do something similar and found that modeling shooters wasn’t really the best way to go about it. It was written by a super bright kid

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Are you able to share what you found was the relationship between wheel surface speed and ball exit velocity?

Interesting! We found that going the opposite way (low friction) worked for us. We got our best results lining our hood with thin PTFE (“Teflon”) sheets. Although the static build up meant reaching into the hood after a bunch of shots was quite the wake-me-up!

I’ve had mixed emotions about these “slip” hoods…. My gut tells me that any amount of slip would create inconsistencies between balls. Adding a top roller was out solution to a higher shot velocity without introducing slip.

I can not deny the logic of that. And perhaps we saw that — we went to our 2020 Week 1 competition as a “behind the wheel of fortune” shooter, but it didn’t take too long for the power cells to get too mangled for that to work effectively due to shot variability<1> However, in 2021 for IR@H (where we were taking care not to mangle our power cells) we found our challenge was really in the horizontal plane.

<1> And also we, among several teams, had target tracking issues due to some extraneous light sources in the venue. With sometimes hilarious results.

I wonder if there was another variable that was leading to less consistency. By making your shooter slicker, it mitigated that variable, leading to a more consistent shoot.

We found this year that a light weight shooter wheel was losing too much inertia as the ball was moving through the hood. We ended adding 5 lbs of weight, and are soon to experiment with 7 lbs. It helped tremendously. Shout out to @tickspe15 for the suggestion.

As for Op’s question, I second the suggestion to determine it experimentally. If not possible, the approximation is a good idea as well.

What do you plan to use this formula for? Are you designing a shooter or trying to get the ball to come out a certain speed via programming? Your tag seems to imply the later.

There were several.

  • We had maybe just a bit too much compression, so IMO the slickness helped there with getting consistent “pitch”. As i mention previously, our bigger challenge was with “yaw”:
  • We discovered that we were not coming out straight and we sometimes had a bit of side-spin. There was just a wee bit too much space in that direction. So, we put the PTFE on the shooter sides as well. then the ball was always centered on the shooter roller and it prevented it from touching just one side and picking up a bit of side-spin.
  • Also our controls team had the time in 2021 to do some great work tuning the turret “PID”. As well as changing how we shot. In 2020, one button spun up the shooter and a second one activated the roller to feed in a ball. If you jumped the gun a bit on the shoot button, (as one might do in excitement), you got a weak shot. In 2021, it was one button and it automatically did the ball feed when the shooter was right at the correct speed for the distance.

Sort of related, for 2021 we quickly realized we could not shoot from the closest zone because we had designed as long range shooter and our shot was just too flat to make it up to the goal from the closest zone. We just used some simple geometry to come up with what the ideal departure angle was for each of the four zones and used that to guide our redesign of the shooter hood to have variable geometry. We considered a completely new continuously variable design, some sort of 4-position design (1 for each zone), but our math indicated that two positions would be sufficient (varying the RPM would be enough to let each position work well for two zones). We again used the board to verify that balls were leaving our new hood as expected in each position. This was time well spent. I think our first run at Interstellar accuracy was something like 42/45 and it didn’t take long to get the desired 45/45.

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This is very interesting. My initial guess would be 1/2 x the speed of the wheel, if it’s a single wheel shooter. Top is stationary, wheel is moving. Ball is rotating against the top and bottom, at what I would initially assume to be 1/2 the speed. Let me know what it actually is if you test it

Of course, assuming no slipping between the wheel and ball

1/2 the surface speed of wheel is going to be a maximum .

My wild guess is most teams are between 20-45% of surface speed.


In our prototyping we found that 30% shot efficiency was pretty standard.

Just eyeballing it, the ball in my video still appears to be traveling at about 30’/sec and I don’t know the shooter speed in that video, but usually it was around 70’/sec.

Thinking more about it, it would have to be a ratio of the circumferences. I think of it like a planetary gear. The sun gear is the shooter wheel, planet gear is the ball, ring gear is the shooter hood/back plate.

The exit velocity (assuming no slippage or loss of energy) is the ratio of the circumferences x surface speed x 1/2

Am I on the right track here?

I envisioned it more as a rolling ball going along the ground.

The bottom surface of the ball is in contact with the ground and has essentially 0 velocity relative to the ground.

If you want the ball to roll at velocity X relative to the ground and the bottom surface is at velocity 0, the top of the ball has to be moving at double the target velocity at the top surface.

Your shooter design is going to set a maximum theoretical velocity - everything else is going to go down from there. The contact time with the wheel is short and the acceleration is high - things are going to slip so your energy imparted to the ball in terms of velocity is going to be less.

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Nope. All that matters is surface speed (to a first approximation). Imagine you had a belt instead of a wheel - would the size of the pulley matter, so long as the speed of the belt is held constant?

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Good point… So would it be surface speed / circumference of ball / 2?

Surface speed per min = circumference of the driving wheel * RPM

How you use that number is up to you…

With a hooded shooter you have to cut it by 2 and then some more based on your empirical results.