You just landed on the planet Olympus IV, famous for its sports competitions. It's a light planet: the average density is one quarter of the Earths average density. It's also a large planet: the diameter is the double of the Earths diameter. The Olympians are giving a high jump competition, and one crew member takes part. You know that on Earth he jumps 1.98 meters. The record at Olympus IV is 3.86 meters. Should you bet your salary on him winning the competition ?

R=radius of Olympus; r= radius of Earth
V=volume of Olympus; v= volume of Earth
M= mass of Olympus; m= mass of earth; w= mass of crewman
F=crewman's weight on Olympus; f= Crewman's weight on Earth

The radius is double that of earth. R=2r

Therefore the volume is 2³ times that of earth. V=2³v=8v

The density is ¼ that of earth. so the total mass is twice that of earth. 
M=¼(8)m=2m

The force of Gravity (the crewman's weight) is half that of earth. 
F = GMw/R² = G(2m)w/4r² = Gmw/2r² =f/2

Barring friction/viscosity issues of air resistance, the graph of thecrewman's height as a function of time during the jump from a flat surface (a Diskworld) within a constant force of gravity would be a parabola described by the equation h=gt²/2=ft²/w  and the maximum height is s²/2g = wv²/2f, where s is the initial velocity.

Since the effect of the variation in the force of gravity is less than that of the air resistance, then these equations are close enough.  If he can jump ws²/2f on Earth, on Olympus he can jump wS²/2F = wS²/f. If V=v, then he can jump twice as high, and just match their record.

The question is does V=v? I'm not sure.

Edited on March 14, 2005, 12:15 pm

Posted by TomM on 2005-03-14 11:21:18