gravity - Why is there an escape velocity?

I've been trying for days, but I just can't understand why escape velocities exist. I've searched the web and even this site, and although I've read many explanations, I haven't been able to truly understand them. Most of the explanations I've seen involve calculus; I only know very little calculus. Could somebody provide a more intuitive explanation?

Here's what I know & understand:

• Escape velocity is the speed at which the kinetic energy equals the gravitational potential energy of an object



• Escape velocity isn't the same as orbital velocity, a satellite never reaches escape velocity, escape velocity is only an initial velocity. And all the other common misconceptions.

What I do not understand, is why an object that has reached escape velocity will never return back to the planet it was launched from. To help you understand where I'm stuck, here's a short "proof":

1. Our whole universe consists of only two bodies. A teapot of mass m, and a planet of mass M. M is many million times larger than m, so the gravitational force acting on the planet is trivial.


2. We launch the teapot from the planet, giving it an initial velocity U (relative to the planet). U > escape velocity for that particular planet.


3. Now, the only force acting on the body (teapot) is the gravitational force from the planet, resulting in a negative acceleration (relative to the planet). The force, and therefore the resulting deceleration, will be decreasing quadratically as distance increases. The negative acceleration will get very close to, but never quite reach, zero.


4. Therefore, the speed of the teapot will never stop decreasing. The teapot will keep decelerating forever.


5. We conclude that at some point, the speed of the teapot will reach zero. The teapot will then fall back to the planet, even though U was greater than the escape velocity.

I suspect that my implication (4) => (5) is false. Somebody please explain why, using as little Calculus as possible. Could this be similar to the Achilles and the Tortoise paradox?

Thanks in advance!

Answer

I don't know if it will help you, but what you are missing is the basic insight of calculus if you want. This lack of understanding generated paradoxes since the time of the Greeks. See "Achilles and the tortoise" on Wikipedia.

The basic point is that you can sum an infinite number of "intervals" (real numbers) and obtain a finite result. For example if you sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$ you get 1 (perfectly finite). The idea is the same with the deceleration. Deceleration reduces the velocity a little bit in a certain time interval, than you are farther away, deceleration becomes smaller and reduces the velocity again but a little less then before, etc. The point is the the sum of all the small reductions of velocity is finite and if this sum is smaller than the initial velocity the velocity will never reach zero and the teapot will always be flying farther away never coming back.

For example if the initial velocity is 2 and the deceleration reduces the velocity in little steps like this $2 -\frac{1}{2} -\frac{1}{4} -\frac{1}{8} -\frac{1}{16}\cdots$ the final velocity is $2-1=1$, still positive! If it started with velocity less than $1$, the velocity would become negative and the teapot will fall back on the planet. I hope it helps your intuition, but study calculus, it is useful ;)