
#1
Dec1113, 01:05 PM

P: 16

I know that for short distances from the earth's surface, x=1/2gt^2+vt+x works fine for finding the time it takes for an object to fall a certain distance ignoring air resistance.
However, what if the distance is many times the earth's radius? The only thing I can think of to start solving this problem is f''(t)=GM/(f(t))^2, but try as I might, I cannot solve that to evaluate the time and object takes to fall, say, a distance, r. How can I accomplish this? 



#2
Dec1113, 01:29 PM

P: 34

I'm assuming you're talking about a body whose mass is much smaller than the Earth's, dropped with zero velocity from an arbitrary distance from the Earth. If so, please check out this article or this one. Basically the body follows a degenerate ellipse trajectory, similar to the trajectory of the moon around the earth. The equations aren't as easy to solve as the mgh potential because they're non linear, but if you want the time in terms of initial position, you write down the energy per unit mass, which is a conserved quantity (and should be negative if you're effectively falling and not escaping the Earth): [itex]E = E_0 = T + U = (dr/dt)^2/2  \mu/r[/itex], solve for dt and integrate: $$ \int_{t_0}^t dt = t  t_0 = \int_{r_0}^r \frac{dr}{\sqrt{2(E_0+\mu /r)}} = \frac{1}{\sqrt{2\mu}}\int_{r_0}^r \sqrt{\frac{r}{1\alpha r}}dr $$ with [itex]\alpha = E_0/\mu > 0[/itex], if I didn't mess up the algebra. The integral may found in integral tables or WolframAlpha, or computed numerically.
For the more general problem where you can have nonzero velocity and a body with a large mass, read into the gravitational twobody problem, or the Kepler problem if you're not interested in large masses. 



#3
Dec1113, 01:40 PM

P: 16

Thank you! This is exactly what I was looking for.




#4
Dec1113, 02:36 PM

P: 34

Time for an object to fall to a larger one
I've added some stuff to my answer, maybe you'd like to check it out :)



Register to reply 
Related Discussions  
Time for Object to Fall Based on Center of Gravity  Engineering, Comp Sci, & Technology Homework  1  
Time measured by freefall observer near object?  Special & General Relativity  7  
Time it takes object in space to fall to Earth's surfac net F on object = Fg of Earth  Introductory Physics Homework  11  
Why does an object fall in the same time as it takes to reach a peak  Introductory Physics Homework  4  
time for an object to fall  General Physics  5 