Prove that gravitational field intensity at any point in equal to acceleration experienced at that point.

Explanation

The gravitational field intensity `vec(E)_(1)` (here after called as gravitational field) at a point which is at a distance r from `m_(1)` is defined as the gravitational force experienced by unit mass placed at that point. It is given by the ratio `(vec(F)_21)/(m_2)` (where `m_(2)` is the mass of the object on which `vec(F)_(21)` acts)
`vec(F)_(21)=-(Gm_(1)m_(2))/(r^2)hat(r)" "...(1)`
Using `vec(E)_(1)=(vec(F)_(21))/(m_2)` (1) we get,
`vec(E)_(1)=-(Gm_1)/(r^2)hat(r)" "...(2)`
`vec(E)_(1)` is a vector quantity that points towards the mass `m_(1)` and is independent of mass `m_(2)`. The value of `m_(2)` is taken to be of unit magnitude. The unit `hat(r)` is along the line between `m_(1)` and the point in question. The field `vec(E)_(1)` is due to the mass `m_(1)`.
In general, the gravitational field intensity due to a mass M at a distance r is given by
`vec(E)=-(GM)/(r^2)hat(r)" "...(3)`
In the region of this gravitational field, a mass 'm' is placed at a point P, when mass 'm' interacts with the field `vec(E)` and experiences an attractive force due to M. The gravitational force experienced by 'm' due to 'M' is given by


`vecF_(m) = m vecE`
Now we can equate this with Newton's second law `vec(F)=mvec(a)`
`mvec(a)=mvec(E)`
`vec(a)=vec(E)`