May I start by saying that I spent a fair bit of time and brain-power on this, and it's pretty late here, so be gentle if I err.
This question requires familiarity with some simple calculus methods and a basic knowledge of some Physics equations - not too hard if you think about it though. Firstly, a labelled diagram which will be referred to down the track:
http://img139.imageshack.us/my.php?image=1ew4.png
Now for some formula derivation:
Now, we are (or should be) aware that
R = -GmMi/x^2
Which can be expressed as
mai = -GmMi/x^2
Some simple rearrangement of this equation gives
a = -GM/x^2 ------> (the m and i cancel in the denominator)
which will be referred to as Equation 1.
Now, at the earth's surface
W = GmM/r^2
but W = mg, so:
mg = GmM/r^2
Again, we can rearrange this slightly to give
gr^2 = GM
which will now be referred to as Equation 2.
Now, substituting Equation 2 into Equation 1, we have
a = -gr^2/x^2.
You should have been taught that acceleration is commonly thought of as the rate of change of velocity, which can be expressed as:
a = d(1/2*v^2)/dx.
So for this question, we have
d(1/2*v^2)/dx = -gr^2/x^2.
Now onto the question itself -- we will need to use integrals. What we need to do is determine at what distance the velocity is zero - at this distance, the rocket will be drawn back to earth due to the effect of gravity (...theoretically). When v = 0, let x = d, and when v = 6500 ms^-1, let x = 40 km + the radius of the earth = 6418000 m.
Now,
http://img61.imageshack.us/my.php?image=2vo7.png (integral-y stuffs)
So,
http://img97.imageshack.us/my.php?image=3el7.png (more integral-y)
If you are not sure of how I went from (
http://img61.imageshack.us/my.php?image=2vo7.png) to (
http://img97.imageshack.us/my.php?image=3el7.png), then you need to tighten up your calc. skills, son!
Equating,
(6500^2/2) – 0 = 9.8 * 6378000^2[(1/6418000) – (1/d)]
(1/6418000) – (1/d) = 5.300 * 10^-8
1/d = 1.028 * 10^-7
d = 9726000 m
d = 9726 km
So the altitude (above the earth’s surface) reached by the rocket = 9726 – 6378 (remember the rocket was fired from the centre of the earth)
= 3348 km.
Note that the above part (from “Equating,” down) was essentially the only work we had to do in this question – the rest was figuring out just how to do that work and what equations etc. to use for it.
Now, if anything there confuses you greatly then perhaps you should pay more attention in your class; for example, you shouldn’t ask me, “What is R? What are G, m and i? Why did you use those equations? What the fuck are integrals?” etc. – you should already know the answers to these questions. Or you could give up the particular class if you’re finding it too difficult.
Alternatively, if you haven’t covered some of the calculus methods or the physics equations/derivations I dealt with, you probably aren’t ready to tackle problems of this nature and should confer with your teacher. Godspeed.