3181: Jumping Frog Radius

Jumping Frog Radius

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Title text: Earth's r_jf is approximately 1.5 light-days, leading to general relativity's successful prediction that all the frogs in the Solar System should be found collected on the surface of the Earth.

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Explanation

This is one of 55 incomplete explanations: This page was created by an CHAMPION PLANET-JUMPING FROG. I have added a bit about the drawing. It is important I think that the planet with the frog has exactly the rjf radius. This means the frog cannot escape but just barely. Is there a physics relation behind the fact that the jumps height seems to be very close to the radius of the planet, i.e. rjf? Also Can someone calculate the size and mass of the largest object from which a champion frog can achieve escape velocity? Are there some named asteroids that are of sow low a mass that it would be possible for frog to jump of? (Of course there are some small enough... but do any of them have real names, like the one named after Randall)? Don't remove this notice too soon. If you can fix this issue, edit the page!

The Schwarzschild radius is essentially the size of a black hole -- the maximum distance from the center where gravity is so strong that light can't escape.

It is part of a solution to Einstein's field equations. It is usually calculated as the following:

r = (2*G*M)/(c²),

where G is the gravitational constant, M is the mass of the object, and c is the speed of light.

If M is the mass of the Earth it would give the Schwarzschild radius for the Earth which is about 9 mm. (If all of Earth mass was pressed into a sphere of a bit less than 2 cm in diameter, it would become a black hole.)

The comic suggests a more useful radius. The Jumping Frog radius, r_jf, which is the size of a "planet" so that its gravity keeps a champion jumping frog from being able to achieve escape velocity. Thus Randall has instead of c, the 299,792,458 m/s speed of light, used a much smaller value of 4.5 m/s, to represent the maximum speed of a jumping frog. It is possible that Randall got that value from this paper, which on page 179 puts an upper limit on the maximum velocity of adult Australian rocket frogs at 4.52 m/s.

The drawing to the right of the formula shows a planet with exactly the radius r_jf. Thus the frog can jump really high compared to the planets size (in this case about as high as the planets radius), before it unavoidably falls back down, since the planet is just small enough to prevent the frog escaping.

The title text points out that the r_jf of the Earth is about 1.5 light days, which is about 7 times the distance to Pluto (compare to the 9 mm Schwarzschild radius). Since Earth's radius is much smaller than this^{[citation needed]}, no frogs will be able to escape, so all frogs that stray into Earth's gravitational well would collect here on Earth. As far as we know, all the frogs in the Solar System are on Earth, so the data apparently matches the theory.

Transcript

[The panel shows a large formula to the left and a small drawing to the right. The formula's right side is drawn above and below the division line:]

r_jf = 2GM/(4.5^m/_s)²

[The drawing to the left shows a very small planet with the radius indicated with a labeled dotted arrow pointing from the center straight up. A frog is shown jumping on the surface. This is indicated with a parabolic dotted line going from a frog sitting on the surface near the top of the planet, up to the frog shown soaring through the air with its limbs stretched out about as high above the surface as the planet's radius. At this point the frog is making a sound. Then the dotted line goes down to about a quarter of the way around the planet where the frog lands making a noise.]

Arrow label: r_jf

Frog: Ribbit

Landing: Plop

[Caption below the panel:]

More practically useful than the Schwartzchild radius, the Jumping Frog Radius is the radius at which an object's gravitational pull is so strong that even a champion jumping frog can't escape.

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Discussion

firstQwertyuiopfromdefly (talk) 05:17, 16 December 2025 (UTC)

Question: Would a correct interpretation be "if a champion jumping frog were to be located just under 1.5 light-days from earth, and if there we're no other gravitational bodies nearby, and if said frog then performed its mightiest jump directly away from earth, then the frog would eventually be overcome by Earth's gravitational field and would eventually land on Earth's surface"? Pgn674 (talk) 06:26, 16 December 2025 (UTC)

I guess that is exactly how it should be interpreted. Or more interesting if it was just outside this radius and somehow could gain exactly 4,5 m/s extra speed then it would escape Earth (if there was anything to push of against that was heavy enough to move basically only the frog forward, then that would change the mass behind the frog so... That was why I wrote gain exactly rather than jump). --Kynde (talk) 07:36, 16 December 2025 (UTC)

or its mightiest jump in any direction (that doesn't cause it to crash through the Earth) since the escape speed is the same in all directions (relevant xkcd:https://what-if.xkcd.com/68/ ) --178.197.223.163 09:21, 16 December 2025 (UTC)

The only two variables are rjf and M, so plotting a 2 axis graph plotting the relationship between M and rjf should be possible. Zabadoh (talk) 08:20, 16 December 2025 (UTC) ^{[You sign after your contribution]}

As frogs usually collect on the surface of worlds ^{[citation needed]}, the *surface* escape velocity is most important. The crossover point for a planet with earth-like density (5515 kg/m³) is 2.6km, above that, the rjf falls below the surface, and the planet can accumulate frogs. Smaller bodies are, however, usually less dense; an interesting borderline candidate is Chicxulub, which had an rjf of 3-4km, and a radius of 5-6km so could have just about held onto its frogs, for a while at least. JeffUK (talk) 10:04, 16 December 2025 (UTC)

It would be interesting to look at the R_jf values of a frog, to consider where new limits are put upon the frog for M-masses that aren't totally dominating the scenario of "frog leaves mass"... 82.132.237.93 11:03, 16 December 2025 (UTC)

I interpreted it as a reference to the Mark Twain short story The Celebrated Jumping Frog of Calaveras County.

Gustaveeiffel314 (talk) 12:25, 16 December
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