# Help with gravitation proof

## Homework Statement

A rocket accelerates to speed v0 near the earth's surface,and then coasts upwards. Show that if
v0 = 2sqrt(gR)
where R is the Earth's radius, then very far from the earth the rockets speed will be
v = sqrt(2gR)

## Homework Equations

The only equations I have thought of that may help are:
Escape Velocity:
v = sqrt(2GM/R)
Gravitational potential energy:
U= -GMm/r

## The Attempt at a Solution

I have attempted using energy conservation and solving for v, however the answer appears far from that required.
This lead me to think that perhaps linear approximation may be required but I am unsure how I would apply (1+x)n = 1+nx for x<<1 in this problem.

tiny-tim
Homework Helper
Welcome to PF!

Hi K29! Welcome to PF! (have a square-root: √ )
A rocket accelerates to speed v0 near the earth's surface,and then coasts upwards. Show that if
v0 = 2sqrt(gR)
where R is the Earth's radius, then very far from the earth the rockets speed will be
v = sqrt(2gR)

The only equations I have thought of that may help are:
Escape Velocity:
v = sqrt(2GM/R)
Gravitational potential energy:
U= -GMm/r

Yes, those equations should do it …

but they use G, not g …

so what is the formula connecting G and g (and R)? Ah yes. Thank you. All I needed to do was manipulate the final answer that I had from enegry conservations a tiny bit and then substitute that formula in

uhm as a new user,... am I meant to reply once I've got an answer??
I like the style of answering questions. Better than just copying an answer. "Help" is the way to go. At least that way I know where I've gone wrong

tiny-tim
Homework Helper
uhm as a new user,... am I meant to reply once I've got an answer??
I like the style of answering questions. Better than just copying an answer. "Help" is the way to go. At least that way I know where I've gone wrong

Hi K29!

Yes, it's friendlier if people reply. Otherwise we tend to assume that they got a complete answer to the same question on another site, and never bothered with the hints from this site. The hints, btw, are the characteristic policy of PF … Greg Bernhardt reckons that members learn better if they have to do most of the work themselves!

See you around! 