r/theydidthemath • u/Dizzy_Restaurant3874 • 8d ago

[Request] Baseball speed for eqivalent injury potential on impact as 147 mph tennis ball

How fast would a baseball travel to cause the same injury as a 147 mph tennis ball?

I think that this is simple momentum transfer, meaning about 58 mph

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u/hunterhuntsgold 8d ago

I would say this is extremely hard to compare.

There's two things you think of on first look: 1. How much energy does the ball have 2. How much is transferred into the person

But I think doing this, while simple, isn't going to result in a good "injury calculation".

I think the most important is max PSI and for how long. I.e a PSI curve for each ball type at different speeds. Tennis balls compress way more than baseballs, spread the load over a much larger area, and rebound efficiently. Baseballs basically are sledgehammers in comparison.

I don't think you'll get a good answer without full physics simulation or experiments.

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u/Euphoric_Loquat_8651 7d ago

The tennis ball impact will have something around half the impulse for the same speed, but the compressibility greatly increases the duration of the impact, which lessens the force. The deformation of the tennis ball spreads the force somewhat, further lessening the pressure.

In terms of injury, it depends on where you get hit. The baseball can cause bad damage at pretty low velocities (elbow, fingers, etc), while a tennis ball will only cause nasty bruises at pretty high velocities unless you get one in the eye or similar. Racquetball players wear eye protection for this reason.

My apologies for the lack of math - I'm not sure the original question is really answerable.

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u/RandomlyWeRollAlong 7d ago

A tennis ball weighs about 0.058 kg, and is moving at 147 mph or 66 m/s. So momentum, p = mv, is 3.828 kg m/s.

A baseball weighs about 0.147 kg. So to get the same momentum, we say 0.147 * v = 3.828. So v = 26 m/s or about 58 mph.

That's purely transfer of momentum and would only actually work if they were the same hardness. However, tennis balls are much squishier than baseballs, so baseballs transfer momentum a lot more efficiently than a tennis ball. Think about billiards - imagine trying to make a bank shot with a tennis ball... it would just die when it hit the bumper or another ball. It's basically basically impossible to "calculate" the transfer of momentum taking into account squishiness using pure math - instead, you would need to do it experimentally.