Space elevator minimum initial speed

[RiotRick]

Homework Statement

In the far future, humans have built a space elevator as a cheap
means of access to space. However before that could be done, a few basic principles had to be
worked out. . .

a)
What is the minimum initial speed (in an Earth-centered inertial reference frame) needed
for an object launching from the Earth surface (r = rE) to escape its gravitational influence
entirely? You can rely on conservation of energy.

b)
Objects can be launched into space from the elevator “just” by moving them up the elevator,
until a certain height, and then releasing them (with zero radial velocity). What is the
height $r_{esc}$ (as measured from the centre of the Earth) the elevator needs to have so that
the released object would ultimately escape the Earth’s gravitational influence? Careful:
the escape speed has not the same value as for the previous question.

Homework Equations

The previous question asked for the initial escape velocity.
$F_G = G*\frac{M*m}{r^2}$

The Attempt at a Solution

a) is solved via conservation of energy

b) Here I don't fully understand the question. The "careful" makes me suspicious. Do I set centrifugal force equal to the gravitational force or does it ask for the tangential velocity? So I'd have to set the tangential velocity equal to the escape velocity
$\omega*r=\sqrt(\frac{2GM}{r})$

 

[andrewkirk]

First work out what the distance R from the Earth's centre must be for a free-falling geostationary satellite, by equating the formula for required centripetal acceleration (to maintain a circular orbit) to that for gravitational acceleration and solving for radius.

A free-falling object that momentarily has a geostationary velocity will escape the Earth if it is beyond the distance R, and will fall to Earth if it is less than that distance (Why?).

So if the top of the elevator is a satellite at distance beyond the distance R, tethered to Earth by the cable, the object can be released as soon as it is materially beyond distance R.

 

[RiotRick]

I should have posted all the questions:

a)
Is there a circular orbit for which an object would hover continuously above the same
point on Earth? What is the radius rGSO of that geostationary orbit? Hint: for a circular
trajectory, the centrifugal force experienced by the object balances the gravitational pull
from the Earth.
b)
The space elevator starts at a base station on Earth at the equator, crosses geostationary
orbit and extends even further out into space. The whole structure revolves
with the Earth, at the same angular velocity. What is the direction of the force experienced
by an object positioned along the elevator below rGSO? Above?
c)
What is the minimum initial speed (in an Earth-centered inertial reference frame) needed
for an object launching from the Earth surface (r = rE) to escape its gravitational influence
entirely? Hints: in the worst case, the object’s speed would approach zero when r ! 1.
You can rely on conservation of energy.
d)
Objects can be launched into space from the elevator “just” by moving them up the elevator,
until a certain height, and then releasing them (with zero radial velocity). What is the
height resc (as measured from the centre of the Earth) the elevator needs to have so that
the released object would ultimately escape the Earth’s gravitational influence? Careful:
the escape speed has not the same value as for the previous question.

My solutions:
b) what you pointed out
a) $r_{GSO}=(frac{GMT^2}{4*\pi^2})^{1/2)$

c) $v=\sqrt(\frac{2GM}{r_E})$

Now how is d) different from a) and b)?

I don't know what's wrong with my LaTex format

 

[andrewkirk]

The answers to (a) and (d) ARE radii, while the answer to (b) is a direction, so we only need to find a difference between (a) and (d). The answer is the same, but not tautologously so. An argument is needed to justify the conclusion that the greatest lower bound of radii at which a release with zero velocity relative to the elevator will lead to escape is the same as the geostationary radius.

I don't know what's wrong with my LaTex format

See the red parenthesis:

r_{GSO}=(frac{GMT^2}{4*\pi^2})^{1/2)

Also, the backslash is missing from before the frac.