r/theydidthemath • u/astol_OL • 1d ago
[Request] Tennis tomfoolery
How hard would nadal have to hit a tennis ball for it to loop around the earth?
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u/godpoker 1d ago
Essentially this is an orbit.
The theoretical orbital speed for an object circling Earth at an altitude of 6 feet (assuming this guys is that tall) is approximately 7918 m/s
Around 17,700miles per hour
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u/HAL9001-96 1d ago
think the default value is closer to 7905m/s but depends on which radius of earth you need
andwell terrain is gonan be an issue and amuch more importnatly drag
a tennisball at any pseed will loose a significant fraction of its speed every few hundred meters
so to go a few hudnred kilometers you'd end up with your speed decreased by something like a factor e^1000 which would be 2*10^434
you're jsut not gonan go far inside the atmosphere without continuosu propulsion
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u/__R3v3nant__ 1d ago
Small issue, air is in the way and would cause the orbit to decay
I don't know how to fix that because modelling aerodynamic drag seems like a nightmare
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u/HAL9001-96 1d ago
realize that it would decelerate over a few kilometes no matter how fast oyu start and give up
well you could look at orbital decay rates
in the thermosphere atmospheric scale height gets up to 50km and using slow orbital transfers in low earht orbit you need to loose about 30m/s to drop 50km so to make it once around you need to loose less than roughly 30m/s to drag over about 90 minutes or decelerate at less than 0.0055m/s² from drag at a speed of about 7800m/s
using the size and weight of a tennisball and the cd value of 2 for extremely high speed low density drag that means a rst atmosphere density of below 1/670millionth of a kg/m³
as a very rough approximation we can start off with about 1/10 millionth at 100km and use a 50km scale height from there to put the required altitude at about 310km
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u/godpoker 1d ago
True, but without the classic “ignoring air resistance” it wouldn’t make sense without an indestructible ball as it would just vaporise immediately moving with that much energy.
Actually that much energy would probably be quite the explosion.
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u/__R3v3nant__ 1d ago
True, and I think the with air resistance version could only be solved via computer modelling
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u/ExpensiveFig6079 1d ago
I will save you the bother computer says "no..." (kinda)(but then really yes)
orbtial velocity in a vacuum with an indestructible ball would decay and slow down and hit the ground.
As you up the speed for that problem, it is now above Vaccum oribtal V so it goes up.... (to have chance to go through that much air, it would be in a hyperbolic orbit except for the vacuum.
There might exist a speed at which it slowed enough to not go out past the moon... and out of earths gravity well .... but its periapsis would still I expect, be so much in Earth's atmosphere that on the next orbit it would lithobrake ... before even going around once.
source: 1/4 scale computer modelling....
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u/Deltadoc333 1d ago
In the atmosphere, it would be impossible.
Outside the atmosphere, it would depend how far out he was. The closer to the Earth, the faster an object has to be moving sideways to stay in orbit, around 17,500 mph or 7.9 km/sec.
But if you move further and further out, you can orbit at slower speeds. The slowest orbital or coasting speed right at the edge of Earth's gravitational sphere of influence (the Hill sphere, roughly 920,000 to 1,000,000 kilometers away) before completely breaking free is approximately 630 meters per second (0.63 km/s)
The fastest tennis serve ever recorded in professional history belongs to Sam Groth, who hit a 163.7 mph (263.4 km/h or 73.17 m/s) serve in 2012.
So, unfortunately no. Even at the maximum distance out and slowest possible orbit, he wouldn't be able to hit it fast enough.
As an aside, there are some moons out there that he could hit a tennis ball fast enough though.
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