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Differential equations and escape velocities

  1. Feb 28, 2015 #1
    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. jcsd
  3. Feb 28, 2015 #2
    Also, I believe we are ignoring air resistance of the bullet since it is not mentioned in the statement.
     
  4. Feb 28, 2015 #3

    Ray Vickson

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    Solving the DE is doing it the hard way; using conservation of energy is doing it the easy way.
     
  5. Feb 28, 2015 #4
    Yes, my first instinct is to use energy. Unfortunately, this is for a de class and it needs to be solved using a de.
     
  6. Feb 28, 2015 #5

    Dick

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    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.
     
  7. Mar 1, 2015 #6
    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.
     
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