# A problem os Newton´s law of gravitation

• Feb 11th 2009, 08:30 PM
cashimingo
A problem os Newton´s law of gravitation
Hi, i have problems in solving this problem, the thing is that i have solve one problem and based on that i have to solve some others, the first one is this:
The force tha gravity exerts on a body of mass m at the surface of the earth is $mg$. In space, however, Newton´s law of gravitation asserts that this force varies inversley as the square of the distance to the earth´s center.If a projectile fired upward from the surface is to keep traveling indefinitely, and if air ressistance is neglected, show that its initial velocity must be at least $\sqrt{2gR}$, where R is the radius of the earth (about 4000 miles). This escape velocity is approximately 7 miles/second or 25,000 miles/hour.Then solved this problem go to the next ones.

In the previous problem, if $v_e$ denotes the escape velocity and $v_0, so that the projectile rises high but does not escape, show that

$h=\frac{(v_0/v_e)^2}{1-(v_0/v_e)^2}R$

is the height it attains before it falls back to earth.
The other problem is
Apply the ideas in the first problem to find the velocity attained by a body falling freely from rest at an initial altitude 3R above the surface of the earth down to the surface. What will be the velocity at the surface if the body falls from an infinite height?