We plan to shoot Sheldon the circus Platyformer out of cannon to a maximum heigh

Question

We plan to shoot Sheldon the circus Platyformer out of cannon to a maximum height of 2000.0 miles above the Earth. What must Sheldon's initial speed be if he is shot straight up? Neglect the effects of air and those of other celestial bodies. Assume that the Earth is a uniform sphere. You may not use the assumption that gravity is independent of altitude! Hint: If the distance of Sheldon from the center of the Earth, r, is always greater than the radius of the Earth, R, then Sheldon's acceleration due to gravity is given by gravity due to Earth = -g R2/r2 where r and R are in meters and g=9.80 m/s2, points away from the center of the Earth.

Explanation / Answer

change in gravitational potential enery = change in kinetic energy

-GMm/(R+h) +GMm/R = 0.5 mv^2     here R = 6400 km and h = 2000 x 1.609 = 3218 km

mgh(1 + h/R) =0.5 mv^2                we can write GM/R^2 = g    where g = acceleration due to gravity

V = 6.480 km/sec

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