Free fall acceleration astronauts problem

In summary, the problem involves a starship orbiting a distant planet of radius R with an altitude h. The astronauts find that the free fall acceleration at their altitude is half the value at the surface of the planet. Using the equation F=GMm/r^2=ma, the solution is found to be h=R(√2-1).
  • #1
bcjochim07
374
0

Homework Statement


A starship is circling a distant planet of radius R. The astronauts find that the free fall acceleration of their altitude is is half the value at the surface of the planet. How far above the surface are they orbiting? The answer should be a multiple of R.


Homework Equations


F=GMm/r^2 = ma

a=g=GM/r^2


The Attempt at a Solution


h is the altitude of orbit

GM/(R+h)^2 = .5* GM/R^2

2GM/(R+h)^2 = GM/R^2

2GM=(GM/R^2)(R+h)^2
2= (R+h)^2 / R^2
2R^2 = (R+h)^2
sqrt 2 * R = R+h
h= R(sqrt2 -1)
 
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  • #2
Bang on!

That is, your are right.
 
  • #3
bcjochim07 said:
GM/(R+h)² = .5* GM/R²

2GM/(R+h)² = GM/R²

2GM=(GM/R²)(R+h)²
2= (R+h)² / R²
2R² = (R+h)²
√2 * R = R+h
h= R(√2 - 1)

Hi bcjochim07!

Yes that's right! :smile:

(Why were you worried about it? :confused:)

But a bit long-winded … you could have cut it down to:

GM/(R+h)² = .5* GM/R²

so 2= (R+h)² / R²

so √2 = (R+h)/R = 1 + h/R

so h= R(√2 - 1). :smile:

Alternatively, start by saying let r be the height above the planet's centre.

Then GM/r² = .5* GM/R²,

so r = R√2, so h= R(√2 - 1). :smile:
 

What is free fall acceleration?

Free fall acceleration is the acceleration an object experiences when falling freely due to the force of gravity. This acceleration is constant and is approximately 9.8 meters per second squared on Earth.

How does free fall acceleration affect astronauts?

Astronauts in orbit around Earth experience free fall acceleration due to the gravitational force of the Earth. This causes them to feel weightless and allows them to float and move around in their spacecraft.

Do astronauts experience free fall acceleration on the moon?

Yes, astronauts on the moon experience free fall acceleration due to the moon's gravitational force. However, the moon's gravitational force is much weaker than Earth's, so the acceleration is not as strong.

Why is free fall acceleration important for astronauts?

Understanding free fall acceleration is important for astronauts because it helps them navigate and move around in space. It also affects how objects behave in space, such as how they orbit around planets.

How is free fall acceleration affected by the distance from Earth?

The further an object is from Earth, the weaker the gravitational force and therefore the weaker the free fall acceleration. This is why astronauts in orbit around the Earth experience a lower acceleration than those on the surface.

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