# Differential equation simplification

1. Dec 8, 2008

### teplotaxl

Hi guys! It's my first post here... I'm in a Computer Science degree, where courses dealing with differential equations are not mandatory (and I have therefore not completed any), and was asked to solve the following problem using the fourth-order Runge-Kutta method, coded in FORTRAN. I obviously have no difficulties with the latter, but I'm COMPLETELY CLUELESS about the mathematical juggling necessary to simplify the equation I will describe below. Any help will be greatly appreciated! :D

1. The problem statement, all variables and given/known data
It is known that all small bodies are attracted towards Earth due to gravitation. To solve the problem, you must simplify the following differential equation and obtain two first-order differential equations. Apply the formulas given by fourth-order Runge-Kutta to find speed y'(t) and position y(t). How long does it take the object to fall?

2. Relevant equations

The equation that describes the movement of an object falling towards Earth is

$$\ddot{y}(t)=-g \frac{R^2}{y^2}$$,

where g is gravitational acceleration, and R is the radius of the Earth.

Take $$g=\frac{32}{5280}$$, $$R=4000$$ and $$H=200000$$ (H is the height from which the object will be dropped).

3. The attempt at a solution
I've tried to integrate both sides of the equation, but cannot obtain a polynomial expression needed to apply RK4.

Cheers!

2. Dec 8, 2008

### Dick

If you need to split it into two first degree equations to apply a numerical method, then let w(t)=y'(t). Your two equations become w'(t)=-g*R^2/y^2 and, uh, y'(t)=w(t).

Last edited: Dec 8, 2008
3. Dec 8, 2008

### teplotaxl

Hahaha I can't believe the simplicity!! I had been trying some exponential stuff... with no result. Thank you very much. I'll try this and post back.

4. Dec 9, 2008

### HallsofIvy

Staff Emeritus
Now that you have two first order equations, run two Runge-Kutta solvers simultaneously, using the values from both at each step.