# Gravity and acceleration

jbmolineux
Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?

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Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?

Not reallly. It doesn't matter how a force is related to distance between the objects, acceleration (by definition) is (distance/second)/second.

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For gravity, inverse square law meas that acceleration is a function of distance, so typically chain rule is used to convert time based acceleration into position based acceleration. Define the two masses as m1 and m2 :

a = dv/dt = v dv/dr = -G (m1 + m2) / r2

For an initial distance r0 and final distance r1, the above equation is integrated twice. The first integration isn't too bad, but the second one is complicated, and you end up with time as a function of initial and final distance, which can't be converted into distance as a function of time.

Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?
The rate at which falling bodies accelerate is the local strength of the gravitational field.

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The rate at which falling bodies accelerate is the local strength of the gravitational field.
For a two body system, this would be the rate of acceleration towards a common center of mass for the two body system (use the common center of mass as the source for a reference frame). Each mass accelerates towards the common center of mass based on the gravitational field of the "other" mass.

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