The solutions on this site are not clear. If someone could explainthem to me or
ID: 1815737 • Letter: T
Question
The solutions on this site are not clear. If someone could explainthem to me or do this problem an alternative way, I would be gladto rate lifesaver!!!Show that the variation of gravity with height can be accounted forapproximately by the following potential energy function:
V=mgz(1-z/re)
in which re is the radius of the Earth. Find the forcegiven by the above potential function. From this find the componentdifferential equations of motion of a projectile under such aforce. If the vertical component of the initial velocity isv0z
Explanation / Answer
Potential energy due to gravity at any point is given by U = GMm/h where, U = Potential energy G = Gravitational constant M = Mass of earth m = Mass of body h = Hight of the body from the center of earth no at a hight z from the surface of earth(radius = re) U = GMm/(re+z) = g*re2*m /(re+z) ..................(g = GM/re2 or, GM =g*re2) = mgre(re/(re+z)) = mgre[1/(1+z/re)] = mgre(1+z/re)-1 as re is very large (6371 km), normally z/re is very small so (1+z/re)-1 = 1-z/re+(z/re)2 + ...... which can be approximated as 1-z/re so, U = mgre(1-z/re)