Question

The gravitational force, F between an object and the Earth is inversely proportional to the square of the distance from the object to the center of the Earth. If anastronautweighs 206 pounds on the surface of the Earth, what will this astronaut welgh 100 miles above the Earth? Assume that the radius of the Earth is 4000 miles(Round off your answer to the nearest pound.)

Answer 1

To the nearest pound, the weight of the astronaut 100 miles above earth is 196 pounds

How to get the weight

We have to solve for the weight using this proportion

`(w_2)/(w_1) = (R^2)/((R+h)^2)`

such that w₁ = weight of the astronaut obn earth surface = 206

w₂ = weight above earth

R = 4000

h = 100

We have to put the values on the formula

`(w_2)/(206) = (4000^2)/((4000+100)^2)`

`(w_2)/(206) = (16000000)/((16810000))`

when we solve the equation above we have

206 * 16000000 = w₂ * 16810000

3296000000 / 16810000 = w₂

w₂ = 196.07

To the nearest pound, the weight of the astronaut 100 miles above earth is 196 pounds

Answer 2

The gravitational force F

`F\propto(1)/(r^2)`

where r = Distance of the object from the center of the earth

Equation

`F=(k)/(r^2)`

and k = proportionality constant

For the astronaut weight = 206 pounds

and radius = 4000 miles

`\begin{gathered} 206=(k)/(4000^2) \\ k=4000^2*206 \\ k=3.296*10^9 \end{gathered}`

If the astronaut is 100 miles above, the radius would be 4100

`\begin{gathered} F=(3.296*10^9)/(4100^2) \\ F=196.07\text{ pounds} \end{gathered}`

The answer would be 196 pounds