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I am not sure how to title this question so apologies if it's inaccurate.

If I throw an object at thrice the escape velocity of earth, what would be its velocity very far away from earth, (at a distance of infinity)?

I tried solving this question by noting that while the velocity (33.6 km/s) is constant, the acceleration is dependent on distance, I don't want to take acceleration as constant.

Acceleration is given by $\frac{GM}{(r+s)^2}$ where $s$ is the displacement of the object,

I thought I should integrate this distance dependent acceleration from $0$ to $\infty$ with respect to $s$ but then I get something which isnt a velocity or acceleration?

How do I solve this problem

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    $\begingroup$ I don't think there is a "ds" in the expression of acceleration. $\endgroup$
    – Y Z
    Commented Jun 20 at 8:30
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    $\begingroup$ Use conservation of energy i.e. KE + PE = constant $\endgroup$
    – John Rennie
    Commented Jun 20 at 8:52
  • $\begingroup$ This may be helpful: en.wikipedia.org/wiki/Hyperbolic_trajectory $\endgroup$
    – PM 2Ring
    Commented Jun 20 at 9:22
  • $\begingroup$ Your object can escape Earth, but it doesn't have enough orbital energy to escape the Sun. $\endgroup$
    – PM 2Ring
    Commented Jun 20 at 9:31

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With physics (and many other sciences as well), most of the time there are multiple approaches to solve a problem.

Of course, you can follow the path that the object describes, and integrate the varying acceleration as a function of distance, but that will be quite a complex maths task.

An easier approach uses conservation of energy: the sum of potential energy plus kinetic energy stays constant over all the path. As this question is labelled "homework-and-exercises", I won't solve the problem for you, but the following questions should bring you on track:

  • What is the potential energy of an object at earth's surface?
  • What is the potential energy of an object at infinity?
  • How much must the kinetic energy of the object change between surface and infinity to compensate for the change in potential energy?
  • What does that mean for its velocity?

Hint for relating velocity at surface level with potential energy: if an object starts with "escape velocity" from the surface, it will slow down to zero at infinity.

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