- #1

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## Homework Statement

a) A mass ##m##, initially at rest, is released from infinity, and is attracted towards a planet of mass ##M##, displacing towards it in a direction parallel to the gravitational field lines present between the masses. Assuming that the only force acting on ##m## is the gravitational force of attraction, and provided ##M>m##, find an expression for the velocity ##V_0## of the mass ##m## at a separation distance of ##x## from ##M## in terms of ##G##,##M## and ##x##.

b) Given that the velocity required for an object to be in circular motion at a distance ##r## is given by ##\sqrt{\frac{GM}{r}}##, determine if ##m## will enter a circular orbit around the planet.

## Homework Equations

##a=\frac{GM}{r^2}##

##\delta PE = -\delta KE##

## The Attempt at a Solution

I tried finding an expression for ##\int_x^∞ a## ## dt## by letting it equal to ##v \frac{dv}{ds}## (since there is no ##t## in the formula for ##a##), but was unable to find an expression for ##\frac{dv}{ds}## as well(s=displacement).

Then I tried using the law of conservation of energy to solve it, and I used the equation ##KE = \frac{GMm}{x} = \frac{1}{2}mv^2## (deriving it from the second relevant equation) and came to the expression ##v = \sqrt{\frac{2GM}{x}}##, but I have my doubts about this.

As to the second part, I'm lost.

Any help is appreciated.