# What if we set the exact equality $u^2=\frac{2GM}{R}$? Escape or not?

Let an object of mass $$m$$ is thrown up with an initial velocity $$u$$ from a spherical gravitating body $$A$$ of mass $$M$$ and radius $$R$$. Let us assume that when it reaches a height $$h$$ its velocity reduces to $$v$$. From energy conservation $$\frac{1}{2}mu^2-\frac{GMm}{R}=\frac{1}{2}mv^2-\frac{GMm}{R+h}.$$ Therefore, final kinetic energy is $$\frac{1}{2}mv^2=\Big(\frac{1}{2}mu^2-\frac{GMm}{R}\Big)+\frac{GMm}{R+h}=E_0+\frac{GMm}{R+h}.$$ Now, if the initial velocity $$u$$ is such that $$u^2\geq\frac{2GM}{R},~\text{or}~E_0\geq 0,$$ the kinetic energy (hence, velocity) will never become zero. It will attain a minimum nonzero value when the second term on the RHS become zero i.e., at $$h\to \infty$$. Beyond this point the potential energy will remain zero but the kinetic energy remains nonzero implying that the body will go higher and higher escaping the gravitational pull of $$A$$.

Now let us analyze what happens if we set the exact equality $$u^2=\frac{2GM}{R}$$. In this case, both the LHS and the RHS of the first equation is zero. As $$h$$ increases $$v$$ decreases. There will come a point at which $$v=0$$ and that point the gravitational potential energy will also be zero to conserve energy. At this point, the body has zero kinetic energy but maximum (zero) potential energy. Why will not the particle return back? For a pendulum in its extreme position has maximum potential energy but zero kinetic energy. It does return back.

I think in case of pendulum it returns back because the force is still nonzero at the extreme position. This will cause the particle to accelerate towards the mean position. But for the present situation, at $$h\to\infty$$, the force vanishes as $$F\sim r^{-2}\to 0$$. So nothing can cause it to accelerate back towards A. So it will float in space. Am I right?

• This is a theoretical spherical cow. You are overthinking a bit. The theoretical spherical cow is that the entire universe consists of two point masses, that this two point mass universe obeys Newtonian physics, and that those point masses can have a relative velocity exactly equal to escape velocity. If that is the case, it will take the point masses an infinite amount of time to eventually come to rest with respect to one another at infinity. – David Hammen May 3 '19 at 14:38
• it just means that the farther away the particle is, the closer the speed will become to zero, asymptotically. – Wolphram jonny May 5 '19 at 0:53

It does not go back because the kinetic energy is zero only when it reaches infinite distance from the origin of the potential, hence the potential energy and the force is also zero.

In practice, far away from for example the Sun, objects will move very slowly and their orbits can be easily perturbed by other objects. This applies to objects in the Oort cloud, from which comets may fall to the Sun when perturbed.

• I have added a paragraph at the end. Do you agree? – mithusengupta123 May 3 '19 at 14:25
• Yes I agree ... – my2cts May 3 '19 at 15:10

There will come a point at which $$v=0$$ and that point the gravitational potential energy will also be zero to conserve energy. At this point, the body has zero kinetic energy but maximum (zero) potential energy. Why will not the particle return back?

That point is never reached. It occurs at $$r=\infty$$. It takes forever to reach that point.

The analogous situation with a pendulum is if you give the pendulum exactly the amount of kinetic energy needed to make the pendulum come to rest when it becomes perfectly inverted. With more kinetic energy than this limit, the pendulum passes through the inverted pendulum point and never coming to a stop. With less kinetic energy than this, the pendulum reaches its maximum height in a finite amount of time. The amount of time needed before the pendulum comes to a rest before returning increases as the amount of energy approaches this limit. And at this limit, the amount of time needed to come to a rest is infinite.

• I think physically, $\infty$ here will mean a very large distance not mathematical infinity. What do you think about my explanation in the last paragraph? – mithusengupta123 May 3 '19 at 14:23
• @mithusengupta123 No, $\infty$ means infinity. At any finite distance, no matter how large, both the velocity of the object and the force on it will be nonzero. – J. Murray May 3 '19 at 15:39