# Escape Velocity and Integrated Acceleration

Q:
I have always had a hard time understanding the concept of escape velocity on a philosophical level. I get that at any finite distance, we can calculate the "current" speed of a "bullet" which had escape velocity when fired away from the earth and those velocities seem to always be greater than 0 no matter what arbitrarily large value for the distance we select. I also get how the concept of a mathematical limit applies to the case in that the amount of gravitational acceleration experienced by the bullet approaches 0 faster than the velocity of the bullet approaches 0. So far so good. My problem is that since gravity has an infinite range, and the universe has an infinite size, and the gravitational force has an infinite amount of time to act, the sum of the acceleration experienced by the bullet should have at least one infinite term and this implies that no matter how fast the bullet leaves the earth, it should always eventually come back. That is, the sum total of the acceleration felt by the bullet over an infinite time span should be an infinite series of (incredibly tiny) finite terms, which should add up to an infinite amount of acceleration....More than enough to stop the bullet. Do you follow? Anyhow, I'd love to stop laying awake at night fearing the big crunch that this line of thinking implies will inevitably occur. Thanks for hearing me out. :)
- Bob Shueey (age 40)
Hastings, NE, USA
A:

Your reasoning makes sense except for one error: " the sum total of the acceleration felt by the bullet over an infinite time span should be an infinite series of (incredibly tiny) finite terms, which should add up to an infinite amount of acceleration." The sums of infinite series of positive terms do not have to come out infinite. Think of, for example, 1/2+1/4+1/8+1/16.... Each new term just takes you half-way toward 1 from the previous value. So the limit is obviously 1, not infinity.

Mike W.

(published on 09/30/2016)

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