Question

Show by algebraic reasoning that your gravitational acceleration toward an object of mass M a distance d away is a = GM/d2 and therefore doesn’t depend on your mass.

Answer 1

the force between two objects due to gravity is given by this formula:
\[F _{gravity}=\frac{ Gm _{1}m _{2} }{ r ^{2} }\]where G is a constant: G=6.67x10^(-11)

m1 and m2 are the masses of the two objects, and r is the distance between the two objects.

according to newton: F=ma, therefore a = f/m. using M for the big mass and m for "your mass", ACCELERATION DUE TO GRAVITY IS GIVEN BY:
\[a=\frac{ F }{ m }=(\frac{ GMm }{ r ^{2} })/m=\frac{ GMm}{ r ^{2}m}=\frac{ GM }{r^{2} }\]

we substitute the formula for gravitational force into the formula for acceleration, and the little m's cancel!

let me know if this needs to be clearer.

Answer 2

gravitational force is \[GmM \div r^{2}\]
force is also\[m \times a\]
equating m cancels out and we are left with \[GM/r^{2}\]
where M is mass of the earth