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

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

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.

Use our online calculator to test the equation. Click the image below for the calculator page. use for heights where h<<Radius of the earth.

This image has an empty alt attribute; its file name is variation-of-g-calculator-1.jpg
The formula for acceleration due to gravity at height h – with derivation
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