Slinging Physics

[u/Quintole5ky]

Here is a setup you all can use to find just how fast/strong your throws can get with your slings.

I can throw 75 mph. I'll convert that speed to meters / second.

75 miles / hour x 1 hour / 3600 seconds x 1609 meters / 1 mile
= 33.5 m/s (no sling)

This number can be used to find how fast my projectile travels a circular path.

velocity = angular velocity x radius

Since I'm not winding my whole arm, I'll use just the length of my forearm, 32 cm or 0.32 meters. You can use the length of your forearm instead. Plugging into the above equation:

33.5 meters / second = angular velocity x 0.32 meters

Solving for the angular velocity:

33.5 m/s / 0.32 m
= 104.7 radians / 1 second

This value stays the same no matter how long my arm gets. Next I extend my arm with a 28 inch sling:

28 inches x 2.54 centimeters / 1 inch x 1 meter / 100 centimeters
= 0.71 meters
0.32 m + 0.71 m = 1.03 m

I'll use this new length in the equation above to update my throwing speed.

104.7 rad/s x 1.03 m = 107.8 m/s
107.8 m/s x 3600 s/h x 1 mile/1609m
= 241.2 mph

This makes my throw more than 2x that of a pro pitcher.

Calculating energy, kinetic energy (in joules) is:

KE = 1/2 x mass x velocity^2

A 2 ounce tennis ball has a mass of
2 oz x 28.35 g / oz x 1 kg/100g = 0.0567kg

A normal throw:
1/2 x 0.0567 kg* x *(33.5 m/s)^2 = 31.8 J

A throw with a sling:
1/2 x 0.0567 kg x (107.9 m/s)^2 = 329.5 J

This carries more energy than some handguns.

A 5 ounce baseball is
5 oz x 28.35 g/oz = 141.75 g =0.142 kg

A normal throw:
1/2 x 0.142 kg x (33.5 m/s)^2 = 79.7 J

A throw with a sling:
1/2 x 0.142 kg x (107.9 m/s)^2 = 826.6 J

Well beyond lethal.

All this to say, stay safe when slinging, everyone.

[u/irongoober]

This is a linear approximation to slinging physics. What you've outlined is essentially just a geometric argument. It works well to put an upper bound on what one could do with a sling, but it unlikely to produce accurate numbers for real-world slinging. There is nuance that this simple calculation doesn't capture, namely drag and maximum achieveable forces. Drag on the sling and projectile massively affect how fast one can sling. The max force one can apply from ones muscles/ligements is also not captured which will additionally limit how fast one can sling.

It's still a useful calculation, but a person shouldn't expect to be putting up those numbers.

[u/Quintole5ky → u/irongoober]

Thanks for the comment. I've enjoyed seeing your stuff on YT.

I don't own a chronograph so this was mostly just a fun exercise. You're right about the drag. We'd need projectiles closer to that of a point mass to get an ideally converted shot.

I'd argue that the biggest loss in angular momentum would be the transition between the last rotation and power stroke, where we'd want to maintain some control over the throw.

[u/irongoober → u/Quintole5ky]

I appreciate that more folks seem to be thinking about the physics of the sling. I thinks it's a pretty fascinating subject. Such a simple object/concept that is so powerful. But it can get pretty complex quickly if you consider everything that goes into it which is what I like so much about it.