1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Final velocity due to gravtiy

  1. Nov 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Assuming the Earth is at rest and alone in the universe what speed would a spaceship arriving at the surface of Earth from an infinitley large distance be if the speed at infinity= 0 and acceleration is caused only by gravity

    2. Relevant equations
    [itex]F= \frac{GmM}{x^2}[/itex]
    Also using the chain rule [itex]a=v\frac{dv}{dx}[/itex]

    3. The attempt at a solution

    well I set ma equal to the force due to gravity so

    [itex]v\frac{dv}{dx}= \frac{GM}{x^2}[/itex]

    seperating the variables gives [itex]vdv=\frac{GM}{x^2}dx[/itex]


    [itex]\int vdv=GM\int x^{-2} dx[/itex]

    would the limits of integration need to be from infinity to the surface of the earth, or am i on the wrong track all together?
  2. jcsd
  3. Nov 1, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You're doing fine. Just do the integral as you suggest.
  4. Nov 1, 2013 #3


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    You're on the right track. But you might need to be careful with signs. [Edit: I see haruspex beat me :smile:]
  5. Nov 1, 2013 #4
    Ok, then doing the indefinite integral gives [itex]v=\sqrt{\frac{2GM}{x}} + C[/itex]

    but i'm not sure about the limits, would it be [itex]\frac{1}{2}v^2=-GM\int_{\infty}^{6370}x^{-2}[/itex]

    where 6370 is the radius of the earth in km?
  6. Nov 1, 2013 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The limit is the radius of the Earth. What units you use are up to you, so long as you are consistent. I would think your value for G assumes m, not km.
    I think you have a sign wrong above. The minus sign should only appear after performing the integral.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted