# Rocket propulsion - Differential equations

1. Oct 8, 2012

### Locoism

1. The problem statement, all variables and given/known data

Suppose a rocket is launched from the surface of the earth with initial velocity
$v_0 = \sqrt(2gR)$, 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

3. The attempt at a solution

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

$\frac{dt}{dx} \frac{dv}{dt} = \frac{dv}{dx} = -\frac{mG}{v (R+x)^2}$

which gives $v(x) = \sqrt(\frac{2mG}{R+x}) + C$

Using $v(0) = \sqrt(2gR)$ gives

$C = \sqrt(2gR) - \sqrt(\frac{2mG}{R})$

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

2. Oct 8, 2012

### voko

g, G, m and R are related. Use the relationship to eliminate m and G from your formula for v(x). That should answer (a).

For (b), you just need to integrate (a).

3. Oct 9, 2012

### HallsofIvy

Staff Emeritus
Problems like this really annoy me! The whole point of a rocket is that, unlike simply throwing a rock or ball in the air, is that a rocket engine stays on during its flight. In this problem we are apparently to assume that is not true.

4. Oct 9, 2012

### Locoism

Lol. But what if we assume the rocket is very big ball thrown by a really strong dude?

As for the actual math...

What? How do I do that??? It seems that whenever I integrate of differentiate they will always remain since they multiply y? The only way I can see they are related is that they appear in the same formula for y''... How can I cancel them?

5. Oct 9, 2012

### voko

mg equals the force of gravity on the surface of the Earth. What is the the force of gravity on the surface of the Earth?

edit: "m" here is not the mass of the Earth as you used it above. It is the mass of some small body, such as the rocket. I suggest you denote the mass of the Earth as M to eliminate confusion.