What are the scalar equations displacement of acceleration obeys the inverse-square law?

tidones0r 2022-09-27 Answered
In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as:
a ( t ) = a 0
v ( t ) = a 0 t + v 0
x ( t ) = 1 2 a 0 t 2 + v 0 t + x 0
My question is how would you formulate a similar equation when acceleration is dependent on the inverse-square of distance from a point, such as Coulomb's Law or Isaac Newton's inverse-square law of universal gravitation?
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Answers (1)

nutnhonyl8
Answered 2022-09-28 Author has 8 answers
Since a = d v d t = d x d t d v d x = v d v d x = d d x ( 1 2 v 2 ), integration gives v 2 = v 2 2 k x . Then t = d x v = d x v 2 2 k x . If you evaluate that integral (hint: substitute x = 2 k v 2 csc 2 θ), you'll have t as a function of x, not the other way round.
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