Use Mach’s restatement of Newtonian mechanics to show that if we define the centre of mass of two particles according to,
⃗ r = (m1 ⃗ r1 + m2 ⃗ r2) / (m1 + m2)
then the center of mass moves according to the equation,
⃗r = [(m1 ⃗u1 + m2 ⃗u2) / (m1 + m2)]t + ⃗r0
where ⃗r0 is a constant vector.
This is a condensed version of the notes that were given in class but everything is there :
Mach's approach was to use Newton’s third law, which is the only law of the three to actually state a law or prescription of nature, as a starting point.From Newton's third law we can get the expression
a1/a2 = k12 and k12 is a constant.
So then Mach’s reformulation of Newton’s mechanics states that for any two of two or more interacting particles, the ratio of their acceleration will be constant. If we apply a Galilean transformation we obtain,
a1'/a2' = k12 = a1/a2
showing the equations are form invariant.
We relate the constant with the inertial mass .
Mach’s version of Newtonian mechanics is free from definitions and has therefore simplified the theory by making reference only to acceleration a geometric quantity. All other standard notion of Newtonian mechanics can be derived from Mach’s restatement.
The Attempt at a Solution
I fully understand Newton's three laws and I get what Mach's restatement. I just can't figure for the life of me how I can use it to get the desired equation. If anyone has an idea of where to start that would be great!