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

F = ma = GMm/d²

m cancels out on each side of the equation, therefore the acceleration is independent of the mass.

To show that the gravitational acceleration toward an object of mass M at a distance d away is given by a = GM/d^2, we can use algebraic reasoning based on Newton's law of universal gravitation.

Newton's law of universal gravitation states that the force of gravity between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Let's consider two objects: one with mass M1 (the object that produces the gravitational force) and the other with mass M2 (the object experiencing the gravitational force). The force of gravity acting on M2 is given by:

F = GM1M2/r^2 ------- (1)

Here, G is the gravitational constant, and r is the distance between the two objects.

Now, let's focus on the object with mass M2 that experiences the gravitational force. According to Newton's second law of motion, the gravitational force on an object is related to its mass and acceleration:

F = M2a ------- (2)

where a is the gravitational acceleration experienced by M2.

To determine the gravitational acceleration, we can equate equations (1) and (2):

GM1M2/r^2 = M2a

Now, we can see that M2 cancels out:

GM1/r^2 = a

Rearranging the equation gives us:

a = GM1/r^2

Finally, we can conclude that the gravitational acceleration, a, toward an object of mass M1 at a distance r away is given by a = GM1/r^2.

Note that in this derivation, the mass of the object experiencing the gravitational force, M2, cancels out, indicating that the gravitational acceleration does not depend on the mass of the object.