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Question

Two bodies having a total mass M (i.e., M = M₁ + M₂) are initially at rest, separated by a distance d, in vacuum, and isolated from all forces except their mutual gravitational attraction.

1. Find the time elapsed for the entire fall from the initial moment to contact.

2. Find the time elapsed from the initial moment to the moment at which the separation is r, such that 0<r<d.

3. Find the time elapsed beginning the instant at which the separation is d/2 and ending when the separation is d/3.

4. Find the fraction of the initial separation that is closed in half of the amount of time required for the entire fall to contact (assuming point masses).

az_lender · Accepted Answer

The gravitational attraction force is G(M1)(M2)/r^2,
so body 1 accelerates at G(M2)/r^2 while body 2 accelerates at G(M1)/r^2.
Hence at any particular r value, their motion is governed by
d2r/dt2 = -G(M1 + M2)/r^2 =>
r^2 d2r/dt2 + G(M1 + M2) = 0.

Although I felt stuck for a little while, your first question is such an obvious one that I was sure of finding a solution on the Internet!  Google "Time That 2 Masses Will Collide Due to Newtonian Gravity" which will take you to physics stack exchange and at least two understandable solutions to question (1).

Steve4Physics · Answer

This should help: https://physics.stackexchange.com/questions/14700/the-time-that-2-masses-will-collide-due-to-newtonian-gravity

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