I would like some help in understanding the derivation of the formula for escape velocity. Why does $t$ approach infinity? And why is $v(t)$ approaching zero? I know that $r(t)$ is infinite because the gravitational field is infinite. But I don't understand why $t$ (time) and velocity would be zero.
I don't think I understand what $v(t)$ is. Is it velocity at a given point in time? And when time increases, you are farther away from the body producing the gravitational field and thus velocity needed for escape is lower? ....
Here is a photo of the derivation:
From the wikipedia page: http://en.wikipedia.org/wiki/Escape_velocity
Answer
You are right in saying that v(t) is the velocity at a given time.
1) The first equation is the equation for conservation of energy for an object in a gravitational field. The left side is the change in kinetic energy and the right side is the change in gravitational potential energy.
2) For escape velocity, we consider what happens a looooong time after the rocket is launched, so we let t go to infinity. For a rocket launched exactly at escape velocity, it has just enough energy to reach a distance infinitely far away from the planet, so r goes to infinity. If the rocket has "just enough energy" to go infinitely far, that means that once it is at r = infinity, its velocity must be 0, since there would be no more leftover energy. Hence v(t) also approaches 0.
3) The article substitutes these values in to the energy equation and solves for the initial velocity, which is the escape velocity.
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