Based on how you phrased this problem, I will assume that x is large compared to r (though it would be much easier if it wasn't), and m is small compared to M.
This brings it down to a question of orbital mechanics, even though this "orbit" is really linear motion....
Okay, that deceptively simple problem had me going for the better part of an hour and involved cracking my diff eq book (something that hasn't happened since long before I graduated), since all my orbital mechanics books are on the shelf in my office right now.
The generic equation for orbital velocity is v = sqrt( 2 G M / r - G M / a ), where a is the semi-major axis. Since your mass is initially at rest at radius (r+x), this means that a = ( r + x ) / 2. (That looks a little odd to me, but I'm going to go with it for now.) Plugging that back into the first equation and setting r = radius of the sphere, that gives you an impact velocity of v = sqrt( 2 G M x / ( r + x) ), assuming I did my math right.
I'm going to stop pulling my hair out now and go do something else now. I hope this helps you.
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timeasmymeasure
Is this an invitation to be nerdy?
Date: 2004-11-25 03:13 am (UTC)This brings it down to a question of orbital mechanics, even though this "orbit" is really linear motion....
Okay, that deceptively simple problem had me going for the better part of an hour and involved cracking my diff eq book (something that hasn't happened since long before I graduated), since all my orbital mechanics books are on the shelf in my office right now.
The generic equation for orbital velocity is v = sqrt( 2 G M / r - G M / a ), where a is the semi-major axis.
Since your mass is initially at rest at radius (r+x), this means that a = ( r + x ) / 2. (That looks a little odd to me, but I'm going to go with it for now.)
Plugging that back into the first equation and setting r = radius of the sphere, that gives you an impact velocity of
v = sqrt( 2 G M x / ( r + x) ), assuming I did my math right.
I'm going to stop pulling my hair out now and go do something else now. I hope this helps you.