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Phys_Boi

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## Homework Statement

Find the total time, t, that an object takes to reach the surface of the Earth from a distance, D, using the Law of Gravitation: $$F_{g} = \frac{GMm}{x^2}$$

R is radius of Earth

D is distance from surface

R+D is total distance from center of masses

****** One Dimension ******

## Homework Equations

F = ma

$$\frac{GMm}{x^2} = ma, a = \frac{GM}{x^2}$$

## The Attempt at a Solution

$$\frac{a}{v} = \frac{dv}{dx}$$

$$a dx = v dv \frac{-GM}{x^{2}} dx = v dv$$

$$-GM\int_{R}^{R+D} \frac{1}{x^2} dx = \int_{0}^{v} v dv$$

$$v = \sqrt{2GM(\frac{1}{R+D} - \frac{1}{R})}$$

$$\frac{dx}{dt} = \sqrt{2GM(\frac{1}{R+D} - \frac{1}{R})}$$

$$dt = \frac{1}{\sqrt{2GM(\frac{1}{R+D} - \frac{1}{R})}} dx$$

$$\int_{0}^{t} dt = \int_{R}^{R+D} \frac{1}{\sqrt{2GM(\frac{1}{R+D} - \frac{1}{R})}} dx$$$$t = \frac{D}{\sqrt{2GM(\frac{1}{R+D} - \frac{1}{R})}}$$

Is this the correct solution to the problem?