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The time rate of change of the position of an object

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Escape Velocity question

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? … 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? …