How would the motion equations look for a free fall of a body from 4000 km, no air friction ?

It is not a motion with a constant acceleration as the earth gravitation is changing inversely with the square of distance between two corps according with Newton's gravitation law.How would the motion equations look for a free fall of a body from 4000 km, no air friction ?If you begin with Newtons laws of motion and gravity, it would be very difficult for you to figure out how to integrate equations of motion. Beter write potential energy in form

PE = GMm/R and use conservation of energy.



Better yet use the following approach:



The falling body obeys the three laws of Kepler.

Write the laws and try to figure out how time depends on altitude t=t(h). No calculus required. It is easier to figure out what will happen if the body has small initial horizontal velocity, the body falling strictly vertically would be the limiting case.



Time dependence on altitude t(h) can be expressed in elemetary functions, but not altitude depedence on time h(t).How would the motion equations look for a free fall of a body from 4000 km, no air friction ?The key here is to remember that the effective mass of the Earth comes from its centre.

Position of body is R, and taking your coordinate system from the centre of the Earth:

R(body) = R(earth's surface) + h

e.g. at initial conditions with height h = 4000km:

R = 6400km + 4000km = 10400km

This is also the distance between their centres of mass, approximating that m(body) %26lt;%26lt; M(Earth)



Newton's second law: F = mR''

Newton's Law of gravitation F = -GMm/R^2 is the only force acting.



Acceleration will be

R'' = -GM/R^2

This is your equation of motion.



This is very difficult to solve for RHow would the motion equations look for a free fall of a body from 4000 km, no air friction ?F= -GMm/(r^2) where G is the Gravitational constant

M is the mass of earth

m is the mass of the body

r is the distance from the centre of earth



therefore

equation of motion is (taking upwards positive)

F=-GMm/(4000*100+d)^2 (if we change km to metre)



where d is the radius of the EarthHow would the motion equations look for a free fall of a body from 4000 km, no air friction ?
You are correct, the motion equations would be based on an acceleration that depended on the distance from the center of the Earth



a=G*me/(re+d)^2



where G is the gravitational constant

me is the mass of the Earth

re is the radius of the Earth

d is the distance above the surface of the earth



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