How Does Rocket Velocity Relate to Escape Velocity and Gravitational Potential?

[K29]

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]

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)?

 

[K29]

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]

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! ​