enterprise 20180401 16:17:05  It is just an order of magnitude calculation. escape velocity=speed of light. You can write this as 2*c=G+MR. Every term is an order of magnitude. Take the equation and apply the logarithm on both sides. For example , M=9*10^24 so just set M=25 in the above equation.  
elgo 20171018 03:15:42  To get the answer exactly you need to know the Schwarzchild radius is at R=2GM. This means for there to be a blackhole, all of the mass has to be inside that radius. To convert G to general relativity units you need to divide it by . When you put in the values for the constants, you\\\\\\\'ll get the right answer of 1cm.  
keradeek 20111001 21:32:38  well, I think the handwaving shown in the official solution is a bunch of crap, cuz energy is gamma*m*v^2, not 1/2. That's like saying the Bohr model of the atom is really informative as to what's going on, cuz it gets the right answer.
keradeek 20111001 21:33:16 
gamma*m*c^2, sorry

 
pam d 20110917 19:06:43  I think down below evanb was the only one who gave a solution that involves general relativity. The Schwarzchild radius is R = 2M (unfortunately knowing this is a combination of what courses you have taken and/or a bit of luck) in c = 1, G = 1 units. You then figure out in SI you need a factor of , simplify the algebra by using G = 7E11 and M = 6E24, and arrive at about 1 cm.
That being said, I really like the plucky handwaving solutions. They demonstrate the type of thinking that will get you some extra points on this test. My favorite is the grape one.  
aqme28 20101112 14:35:17  This is a pretty poor solution, but in this case it works:
I realized the units of G are m^3/(kg*s^2)
Multiply G*M/R and you get units of (m/s)^2.
So by dimensional analysis I figured v is about the square root of GM/R
I got 4 millimeters, which is close enough to a centimeter.  
1234 20100903 11:34:19  Set escape velocity = c:
Escape velocity = [(2*G*M)^(1/2)]/r = c. Solve for r.
x3818919209145 20101111 06:30:12 
Doesn't the r have to go inside the square root?

 
daverbeans 20081106 12:49:13  A really quick and dirty way to do this is to consider the radius of a "stellar mass" black hole, which is about 10 solar masses. These are around 30km in radius. The sun is approximately earth mass. A stellar mass black hole would then be earth masses. so 30km/ is on the order of .01m or 1cm.
Baharmajorana 20140916 01:46:35 
I like your solution, but we have to memorize the radius and mass of black hole!! I think the GREPHYSICS's solution is better.

 
ivalmian 20080325 19:42:17  Um, well, the reason it's called the Schwarzschild radius is that it comes from Schwarzschild's solution of Einstein's equations. Take a look:
http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution  
bat_pesso 20071031 13:50:31  i don't think relativity is such an issue here, the escape velocity depends only on the mass of the earth, not on the mass of the particle.
thus 0.5c^2 is still GM/R  
Jeremy 20071018 13:17:00  Not having seen such hardcore hand waving before, I would not feel comfortable with this solution even if I had thought of it. Looking in my books afterwards, I found something that's pretty cool: you can get from dimensional analysis! (Of course you have to pick the right parameters.) And check this out, , so unless the proportionality factor is several orders of magnitude, the closest answer is (C).
Moral: If thou needeth formulae, consulteth dimensional analysis. I think I gave up on this problem too quickly, when I could have at least submitted an educated guess.  
chri5tina 20061128 05:24:22  So in all this handwaving, did he explain the 1/2?
chri5tina 20061128 05:32:11 
well, I answered my own stupid question, however I'm hitting a bug when I try to edit my above post. I get this error when I try to edit:
omg are you trying to edit someone else's comment, you phisher?! (if you're innocent, then you ought to remember to login.)
when I am already logged in. and I'm not sure where to submit bugs.

 
scottopoly 20061103 19:55:34  I once heard that if the Earth were compressed into a black hole, it would be about the size of a grape. I think that is a much faster solution. ;)
Nebula 20150916 22:29:35 
I heard marble, but still works!

 
ben 20061103 15:21:30  i've generally seen the schwarzchild radius derived from the escape velocity as shown in
http://scienceworld.wolfram.com/physics/SchwarzschildRadius.html  
pablojm 20061028 15:30:36  This is a good argument to convince someone who doesn't know physics, but how do you know it works...?
evanb 20080701 17:22:24 
Unfortunately, the way would know it works is by doing the general relativity. You can deduce that in the Schwarzschild metric for a point mass, any timelike or null (lightlike) trajectory (ie. any trajectory that a particle might take) that starts within R = 2M will ever wind up outside of R = 2M ( G = c = 1).
Ununfortunately, GR reproduces lots of "Newtonian" results. You can even do most cosmology with Newtonian math. Why does that work? Because someone did it fullygeneralrelativistically and found that the math wound up the same...

 