# The formula for acceleration due to gravity at height h – with derivation

Last updated on April 19th, 2023 at 04:59 pm

In this post, we will discuss and derive **the formula for the acceleration due to gravity at height h** (where h<<Radius of the earth).

As height or altitude from the earth’s surface increases the value of g falls. Now we are to discuss and derive the equation of g that describes this change in g with the increase in height.

**The formula for the acceleration due to gravity at height h**

The

formula for the acceleration due to gravity at height h(where h<<R) is expressed by the formula:g1 = g (1 – 2h/R).

Here g1 is the acceleration due to gravity at height h and R is the radius of the earth.

g denotes acceleration due to gravity on the earth’s surface.

**For example, considering g = 9.8 m/s^2 on the earth’s surface, g1 at a height of 1000 meters from the surface of the earth becomes 9.7969 m/s^2.*** [ ***check with online calculator*** ]*

**Derive the Formula for acceleration due to gravity at height h** | Derivation of g at height h

This section covers the variation of g with altitude. At a height of h from the surface of the earth, the gravitational force on an object of mass m is

F = GMm/(R+h)^{2}

Here (R + h) is the distance between the object and the center of the earth.

Say at that height h, the gravitational acceleration is g1.

So we can write, mg1 = GMm / (R+h)^{2}

=> g1 = GM/(R+h)^{2} _________________ (1)

Now we know on the surface of the earth, it is

g = GM / R^{2} [ see the proof: **equation of g on earth’s surface** ]

Taking the ratio of these 2,

g1/g = R^{2} /(R+h)^{2}

=> g1/g= 1/(1 + h/R)^{2} = (1 + h/R)^{-2} = (1 – 2h/R) *[ with the apprroximation that h<<R]*

so,** g1/g = (1 – 2h/R)**

is represented with this equation:The Formula for acceleration due to gravity at height h=> g1 = g (1 – 2h/R) ______(2)(whereh<<R)

g1 is acceleration due to gravity at height h.

You can **use our online calculator to test the equation for acceleration due to gravity at height h. **

Click the image below for the calculator page.Please use it for heights where h<<Radius of the earth.