- #1

- 81

- 0

## Homework Statement

Suppose a rocket is launched from the surface of the earth with initial velocity

[itex] v_0 = \sqrt(2gR) [/itex], the escape velocity.

a) Find an expression for the velocity in terms of the distance x from the surface of the earth (ignore air resistance)

b) Find the time required for the rocket to go 240,000 miles. Assume R = 4000 miles, and g = 78,545 miles/h2

## The Attempt at a Solution

So I figure we use [itex] \frac{dv}{dt} = -\frac{mG}{(R+x)^2} [/itex] and multiply by [itex]\frac{dt}{dx} = \frac{1}{v}[/itex] to get

[itex] \frac{dt}{dx} \frac{dv}{dt} = \frac{dv}{dx} = -\frac{mG}{v (R+x)^2} [/itex]

which gives [itex] v(x) = \sqrt(\frac{2mG}{R+x}) + C [/itex]

Using [itex] v(0) = \sqrt(2gR) [/itex] gives

[itex] C = \sqrt(2gR) - \sqrt(\frac{2mG}{R}) [/itex]

I don't know where to go from here....