Differential equations and escape velocities

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1. Feb 28, 2015

BrettJimison

1. The problem statement, all variables and given/known data
Good day all! I'm stumped on a question:

If I fire a bullet straight up what will be the initial velocity such that the bullet doesn't come back down?
I need to model a differential equation (it will be first order) some how!
Also, Gravity is not constant, but rather, the acceleration due to gravity dv/dt is -k/r^2 where k is a positive constant and r is the distance to the center of the earth (4000 mi)
2. Relevant equations
Also, dv/dt=(dv/dr)v

3. The attempt at a solution
My teacher said something about at the point where the initial velocity is great enough to over come gravity, the root in the de will become complex. That's all I know. Any help would be appreciated!
So far I know:

m(dv/dt)=-mg and g=-k/r^2
I can find a de for v(r) easily since the eqn is seperable, but I'm not sure what to do with it....
Also, the problem gives g at the surface of earth as -32 ft/s^2, and r in miles, so unfortunately we aren't using metric here.

Thanks!!!

2. Feb 28, 2015

BrettJimison

Also, I believe we are ignoring air resistance of the bullet since it is not mentioned in the statement.

3. Feb 28, 2015

Ray Vickson

Solving the DE is doing it the hard way; using conservation of energy is doing it the easy way.

4. Feb 28, 2015

BrettJimison

Yes, my first instinct is to use energy. Unfortunately, this is for a de class and it needs to be solved using a de.

5. Feb 28, 2015

Dick

You could use de's to prove conservation of energy holds first. If dv/dt=-k/r^2 then (dv/dr)(dr/dt)=-k/r^2. dr/dt=v. So v*dv/dr=-k/r^2. Integrate it.

6. Mar 1, 2015

BrettJimison

Hello dick, thanks for the response. I solved the de using a similar method you mentioned above. I checked my solution using a work integral and it was .001 off of the "accepted value". It involved solving the de and finding the critical values and setting a part of the velocity function derived greater than the critical value (so I wouldn't get complex numbers). I'm sure it could have been done many ways. Also, I checked my answer by a simple energy equation as Ray eluded to.