If the initial speed is not too great it is an arc of a sinusoid.Originally Posted by ThePerfectHacker
RonL
Assuming Newtonian gravity (which should suffice) for an object launched from the surface of the Earth we get the equationOriginally Posted by ThePerfectHacker
$\displaystyle 0 = (1/2)mv^2 - (1/2)mv_0^2 - \frac{GmM}{R+y} + \frac{GmM}{R}$
from energy conservation, where v0 is the initial (launch) speed, M is the mass of the Earth, R is the radius of the Earth, G is the Universal Gravitation constant, m is the mass of the object, and y is the height above the Earth. (Note: $\displaystyle g = \frac{GM}{R^2}$) Or we could use Newton's 2nd:
$\displaystyle \frac{GmM}{(R+y)^2} = ma$ (Positive downward.)
Neither equation is trivial to solve for y(t). I can't find my notes on how I did it, but I seem to recall solving the energy equation for y(t) at some point in the past. However, I don't recall getting any great inspiration from it either.
-Dan
The more I stare at the energy equation, the more I doubt I solved it. I may have solved a first or second approximation where y is small.
Thank you, though I can't claim credit...most of it came from one of my textbooks.Originally Posted by Quick
I was thinking that if you really wanted to do the project you could still model the equations even if you didn't have the derivations. The quadratic resistance case is a pretty ugly (oxymoron?) equation but easy enough to program on a fancy calculator, or in Excel if you wanted to make a graph.
-Dan
$\displaystyle 0 = (1/2)mv^2 - (1/2)mv_0^2 - \frac{GmM}{R+y} + \frac{GmM}{R}$
Note that this is the first integral of the Newton's 2nd equation in the previous post where this appears. (5 posts ago)
Solve this equation for v:
$\displaystyle v = \pm \sqrt{\frac{2GM}{R+y}+\left ( v_0^2-\frac{2GM}{R} \right ) }$
v is positive for an object moving upward and negative for moving downward, so we can split the solution into two pieces for upward and downward motion.
For simplicity, take $\displaystyle a = 2GM$ (NOT the acceleration!) and $\displaystyle b = v_0^2-\frac{2GM}{R}$. This equation is then
$\displaystyle v = \pm \frac{dy}{dt} = \pm \sqrt{\frac{a}{R+y}+b }$
Take the first approximation to this equation where y is considered to be small. (ie. keep y to first order, y << R.) This equation becomes an approximate linear first order equation:
$\displaystyle \frac{dy}{dt} \pm \frac{a}{2R^2 \sqrt{a+\frac{b}{R}}} y = \pm \sqrt{\frac{a}{R}+b}$
Letting $\displaystyle c = \frac{a}{2R^2 \sqrt{a+\frac{b}{R}}}$ and $\displaystyle d = \sqrt{\frac{a}{R}+b}$ we see that
$\displaystyle \frac{dy}{dt} \pm cy = \pm d$
has the solution
(*) $\displaystyle y(t) = Ae^{Bt}+D$
Consider now the upward motion solution. (Use the "+" signs.) Inserting this into equation * we get that $\displaystyle B = -c$ and $\displaystyle D = \frac{d}{c}$, so
$\displaystyle y(t) = Ae^{-ct} + \frac{d}{c}$.
Now $\displaystyle v = \frac{dy}{dt}(0) = v_0$. Thus $\displaystyle A = - \frac{v_0}{c}$.
$\displaystyle y(t) = - \frac{v_0}{c}e^{-ct} + \frac{d}{c}$.
Back-subbing to get to the original variables:
$\displaystyle y(t) = - \frac{2R^2v_0 \sqrt{a+\frac{b}{R}}}{a} exp \left [ -\frac{a}{2R^2 \sqrt{a+\frac{b}{R}}}t \right ] $ + $\displaystyle \frac{2R^2 \sqrt{\frac{a}{R}+b} \sqrt{a+\frac{b}{R}}}{a}$
$\displaystyle y(t) = - \frac{R^2v_0 \sqrt{2GM+\frac{v_0^2-\frac{2GM}{R}}{R}}}{GM} $ $\displaystyle exp \left [ -\frac{GM}{R^2 \sqrt{2GM+\frac{v_0^2-\frac{2GM}{R}}{R}}}t \right ] $ + $\displaystyle \frac{R^2v_0 \sqrt{2GM+\frac{v_0^2-\frac{2GM}{R}}{R}}}{GM}$
$\displaystyle y(t) = \frac{Rv_0 \sqrt{2GMR^2+Rv_0^2-2GM}}{GM}$ $\displaystyle \left ( 1 - exp \left [ -\frac{GM}{R \sqrt{2GMR^2+Rv_0^2-2GM}}t \right ] \right )$
To get the rest of the motion, solve y(T) for maximum height, then use the solution to the equation $\displaystyle \frac{dy}{dt} - \frac{a}{2R^2 \sqrt{a+\frac{b}{R}}} y = - \sqrt{\frac{a}{R}+b}$ with the initial condition $\displaystyle \frac{dy}{dt}(T) = 0$. (Or simply consider that the equation for y(t) for the whole motion will be symmetric about t = T.)
As I mentioned previously, it doesn't give me any great inspirations, though it's a nice exercise in applied differential equations.
-Dan
I was just thinking. I did this analysis a few years ago and it never occured to me until now that the solution needs to be forced to be continuous about the time t = T. That could add some interesting complications in solving for the downward motion equation. I'm not going to bother with it. Consider this as a problem "for the interested student."