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Provided that the Solar System consists only of Sol, a planet, and the planet's moon....

How large can my planet be so that it has low enough gravity that teenage human girls trained for jumping, are able to VERTICALLY stand jump as high as their height, and have geological activity that produces earthquakes and volcanoes for at least 5 billion years?

I also want my Earth-like enough so humans might still live there with the ability to make and maintain society & technology like our current society...

Here's some help?

Breathable atmosphere in low gravity

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    $\begingroup$ If you aim to write a story, Worldbuilding might be better suited for this question. $\endgroup$
    – Kyle Kanos
    Commented Apr 7, 2015 at 2:33
  • $\begingroup$ High jump? Standing vertical jump? $\endgroup$
    – DJohnM
    Commented Apr 7, 2015 at 2:48
  • $\begingroup$ Size doesn't matter unless you specify density. I thought the junior female high jump record was pretty close to teenage girl height even for Terra... (1.96 m) $\endgroup$
    – Floris
    Commented Apr 7, 2015 at 8:23
  • $\begingroup$ @Floris - that is the correct answer. Earth-sized. $\endgroup$
    – ProfRob
    Commented Apr 7, 2015 at 10:06
  • $\begingroup$ @Floris So... high jumps are taller than standing vertical jumps due to the running start providing more energy to the jump? $\endgroup$
    – Malady
    Commented Apr 7, 2015 at 12:26

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In general the acceleration of gravity at the surface of a planet depends on both its radius and its mass (density times volume):

$$g=\frac{GM}{R^2}= \frac43\pi G\rho R$$

For a given amount of work done by the athlete ($F\Delta x$), height jumped scales roughly with $g$.

A vertical jump looks like this:

enter image description here

From a standing start you jump up and measure the difference between your reach and the highest point you touch. Really good male basketball players can jump over 1 m this way - and much more with a runup, using the ball as a counterweight.

Assume teenage girls can do 50 cm on earth, then you would need a planet (same density as earth) that is about 1/3 the radius of Earth (with same density) - about 2000 km radius. Gravity would be 3 m/s2 at the surface.

If you don't like my assumptions, tweak them and use the same equation.

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  • $\begingroup$ It will be tough to hold onto an atmosphere when gravity is lower.... Also the core might cool down and become solid (larger surface area to volume ratio so the heat of radioactive fission in the core gets out more easily). This might affect the magnetic field and thus the protection from solar wind... Earth is "just the right size". Maybe your girls need pogo sticks or genetically modified legs. $\endgroup$
    – Floris
    Commented Apr 7, 2015 at 15:14
  • $\begingroup$ So... now the question is how to have low gravity and Earth-like conditions... with normal humans... I could say altered humans, but I don't want to... Should I use Worldbuilding instead? Mmm... I didn't notice that you answered... For future readers, Floris was answering my point that I want humans to live on the planet... $\endgroup$
    – Malady
    Commented Apr 7, 2015 at 15:17

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