High School Physics

# 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 / R2 [ see the proof: equation of g on earth’s surface ]

Taking the ratio of these 2,
g1/g = R2 /(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)

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