# Recursive Projectile To Explicit Formula

1. Nov 25, 2009

### TheMaster...

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

If a projectile orbits around a planet and lands after x cycles of the equation:
at a final radius of C. What is the necessary v?

The only variable that isn't known at the time of solving is v.

Basically I think I need to solve in terms of v but in order to do that I need to make the recursive formula into an explicit one but I can't figure out how.

2. Relevant equations

3. The attempt at a solution
I tried writing down the sequence to find a simpler pattern with no luck. Especially since even that seems to change drastically if I change the constants such as x (the number of cycles). I tried solving it as if it were a normal equation but couldn't figure out how to get rid of the radius[n+1] term to do that and tried setting it to C (the target radius) to radius[n] (so I could ignore it) and to 0 (seemed a better way of ignoring) all with no success seemingly. Setting to radius[n] led to a cubic explicit equation but that doesn't make as much sense as the quadratic equation from setting it to C since we are dealing with motion that is similar to a conic section which is where I derived the original recursive equation from. So how do I start, or how do I make it explicit so I can solve the rest of it.

This isn't a Real homework problem so much as a problem that I desperately need a solution to and couldn't find the right section.

Pointers or solutions are soooo welcome. Thanx :)

2. Nov 25, 2009

### HallsofIvy

Staff Emeritus
I am a bit confused by the fractions on the right side. Is that
$$r_n- \frac{Mv^2}{r_n}\frac{GP}{r_n^2}$$?

Why not just
$$r_n- \frac{MGP v^2}{r_n^3}$$?

3. Nov 25, 2009

### TheMaster...

I'm sorry there should actually be a - inbetween the radius and G not a *. Thanks for taking the time to reply though.

$$r_n_+_1 = r_n- \frac{Mv^2}{r_n}-\frac{GP}{r_n^2}$$

(I stole your LaTex by the way)