Question

In the development of the standard atmospheric model, the variation of the acceleration of gravity with altitude is ignored; that is, the acceleration of gravity is approximated as constant with altitude. Using the Newtonian gravitational model, compute the acceleration of gravity at sea level, 50,000 ft, and 100,000 ft. Based on these computations, describe the accuracy of this approximation.

Answer 1

The accuracy percentage for 50000 ft is 0.47%.

The accuracy percentage for 100000 ft is 0.92%.

Step-by-step explanation:

Given that,

Height = 50000 ft and 100000 ft

We need to calculate the new acceleration due to gravity

Using newtonian gravitational formula

`g'=g*(1-(h)/(R))^(-2)`

Where, g = acceleration due to gravity

h = height

R = radius

Put the value into the formula

`g'=9.81*(1-(50000)/(20.9*10^(6)))^(-2)`

`g'=9.857`

We need to find the accuracy percentage

Using formula for accuracy

`accuracy\ percentage=(g'-g)/(g)*100`

Put the value into the formula

`accuracy\ percentage=(9.857-9.81)/(9.81)`

`accuracy\ percentage=0.47\%`

Now, For other height,

We need to calculate the new acceleration due to gravity

Using newtonian gravitational formula

`g'=g*(1-(h)/(R))^(-2)`

Put the value into the formula

`g'=9.81*(1-(100000)/(20.9*10^(6)))^(-2)`

`g'=9.90`

We need to find the accuracy percentage

Using formula for accuracy

`accuracy\ percentage=(g'-g)/(g)*100`

Put the value into the formula

`accuracy\ percentage=(9.90-9.81)/(9.81)`

`accuracy\ percentage=0.92\%`

Hence, The accuracy percentage for 50000 ft is 0.47%.

The accuracy percentage for 100000 ft is 0.92%.