The Velocity of an Object as it goes to Infinity after escaping a Gravitational field

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The escape velocity is the speed at which the kinetic energy is equal to the gravitational potential energy.

At a distance $r$ from the centre of the planet the gravitational potential energy is:

$$ V = -\frac{GMm}{r} $$

Note that it is negative because we take the GPE at infinity to be zero. As you move away from the planet you GPE increases from a negative starting point up towards zero.

Your total energy is the sum of your kinetic and potential energy, and total energy is conserved. That means the increase in gravitational potential energy must come from a decrease in the kinetic energy, or to put this more simply as you move away from the planet you slow down.

If your initial kinetic energy is exactly equal to the initial potential energy then all of the kinetic energy gets converted into potential energy as you move away to infinity so you end up at infinity with a speed equal to zero. This is what we mean by the escape velocity i.e. the speed that just gets you to infinity. To calculate the escape velocity we simply set the KE and GPE equal to get:

$$ \tfrac{1}{2}m v_e{}^2 = \frac{GMm}{r} $$

and rearranging gives:

$$ v_e = \sqrt{\frac{2GM}{r}} $$

Now suppose you start at some speed, $v_i$, greater than $v_e$. That means your initial kinetic energy is greater than the initial potential energy so after escaping to infinity you have some kinetic energy left over:

$$ KE_\infty = \tfrac{1}{2} m v_i{}^2 - \frac{GMm}{r} $$

To calculate the final velocity, $v_f$, you simply set:

$$ KE_\infty = \tfrac{1}{2} m v_f{}^2 $$

And you can now solve this to calculate $v_f$.

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Updated on February 03, 2020

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  • ArnavT
    ArnavT almost 4 years

    If there is an object, launched from the surface of a planet with twice the escape velocity (v) that is necessary to escape the planet's gravitational field. The question is, when the object approaches an infinite distance away from the planet, what is the velocity?

    I got this question on a test and thought the velocity would be v, thinking that the object will "lose" half of its velocity escaping the planet's gravitational field, and then continue on with v since the potential energy would go to 0. My friend put that the velocity at an infinite distance would be 0. I am not quite sure why though. Is he correct, am I correct, or is there a different answer?