I have the formula for a rocket's escape velocity from earth, $V$ being velocity, $v$ being initial velocity, and $r$ being the distance between the rocket and the center of the earth.
$$V = \sqrt{\frac{192000}{r}+v^2-48}$$
I am trying to find the value of $v$ for which an infinite limit for $r$ is obtained as $V$ approaches zero, this value of $v$ being the escape velocity for earth.
I have solved for $v$ (with $V$ being $0$), as $v = \sqrt{48-\frac{192000}{r}}$, but do not know how to continue solving the problem. I thought setting it up as the limit of the square root of $48-\frac{192000}{r}$ as $r$ approaches infinity (to give $v$), but that doesn't seem right.
Best Answer
Let $r\to\infty$. Note that $\dfrac{192000}{r}\,$ approaches $0$. Thus since $$V=\sqrt{\frac{192000}{r}+v^2-48},$$ $V$ approaches $\sqrt{v^2-48}$. If we want $V$ to approach $0$, we want $v^2-48=0$. (Presumably we are measuring velocity in miles per second.)