Escape Velocity [Math Question Seeking Explanation]

Factual Questions
1

Was working on college homework tonight, when I ran into this problem and couldn’t figure out HOW to solve it. The answer was in the book (And I knew it offhand), but I was curious:

Given the following problem here [Tinypic image of the problem]

How do you solve it to get the answer? Normally I’d just move on with life and leave it for another day, but this time I’m curious as to the math behind the calculation. Probably something stupidly obvious that I’m missing, but if anyone could shed some light on it, that would be great.

Not sure how you guys feel on math threads, so this will definitely be the only one I start. If its a problem, I apologize. Thanks.

2

This is calculus-based intro physics, I assume. This problem is really just an application of the work-energy theorem. Initially, the kinetic energy of the projectile is 1/2 m v[sup]2[/sup]. The work done on the projectile as it goes to infinity is given by the integral of the gravitational force as y runs from 0 to infinity. Finally, we know that

(final kinetic energy) - (initial kinetic energy) = (work done by gravity)

The final kinetic energy (when the particle is at infinity) is zero, since the velocity goes to zero as the particle goes to infinity (see the hint.) This should give you an equation in terms of v, R, and g[sub]0[/sub] (the mass of the projectile should cancel from both sides), and you can then solve this equation for v.

Math threads are A-OK in these parts, but homework threads are somewhat frowned upon, which is why I haven’t filled in all the details above.

3

Thanks for the help. At this point, the homework problem is over and done with, which is why I asked, I just wanted to know the math behind it so I could actually understand.

4

No problem. Upon closer examination, there’s actually an error in the problem — the acceleration due to gravity should be inversely proportional to (R + x)[sup]2[/sup], not (R[sup]2[/sup] + x[sup]2[/sup]). If you actually do the calculation with the acceleration given, the answer ends up being closer to 14.0 km/s instead of 11.2 km/s.

5

And doing the math using that corrected version actually gives me the correct answer the way I integrate and solve. Thanks for pointing that out.

6

If this were a physics problem, the answer could be written down directly using conservation of energy and the well known expression for gravitational potential energy. Since this is a math class, you just have to do the simple integral that relates gravitational force to the potential energy. The numerical answer given in the link is correct.