Help Proving Minimum Speed for Earth Orbit

  • Thread starter decamij
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In summary, the equation for the minimum speed required to maintain orbit around the Earth is v = 2.00x10^7 (root)r. To prove this equation, it is necessary to show that the centrifugal force is equal to the gravitational force, which can be represented by the equation v = GmE (root)r. This can be simplified to v = sqrt(GM/r).
  • #1
decamij
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How can i mathematically prove the following equation:
I have to prove that the minimum speed required to maintain orbit around the Earth is (given the mass of the Earth and universal gravitation constant)

v = 2.00x10^7
(root)r

I have to basically prove this equation:

v = GmE
r

P.S. the whole equation to the above is square rooted, and the r should be UNDER GmE).
 
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  • #2
In order to mantain the equilibrium of the body the centrifugal force has to be equal to the gravitational one:

[tex] m\frac{v^2}{r}=\frac{GMm}{r^2}[/tex]

[tex]v=\sqrt{\frac{GM}{r}}[/tex]
 
  • #3
So if this were a question on an assignment out of 6 marks, all i would have to is show the relationship between the two equations (it would be a pretty short proof). Thanx a lot
 

1. What is the minimum speed needed for an object to orbit the Earth?

The minimum speed needed for an object to orbit the Earth is approximately 17,500 miles per hour, or 28,200 kilometers per hour. This speed is known as the orbital velocity and is required for an object to counteract the pull of Earth's gravity and maintain a stable orbit.

2. How is the minimum speed for Earth orbit calculated?

The minimum speed for Earth orbit is calculated using the formula v = √(GM/R), where v is the orbital velocity, G is the gravitational constant, M is the mass of the Earth, and R is the distance from the center of the Earth to the object. This formula takes into account the mass of the Earth and the distance from the center, which both affect the strength of Earth's gravitational pull.

3. Can an object have a lower speed and still orbit the Earth?

No, an object cannot maintain a stable orbit with a speed lower than the minimum speed required. If an object's speed is too low, it will eventually fall back to Earth due to the pull of gravity. In order to maintain a stable orbit, an object's speed must be equal to or greater than the minimum orbital velocity.

4. What factors can affect the minimum speed for Earth orbit?

The main factors that can affect the minimum speed for Earth orbit are the mass of the Earth and the distance from the center of the Earth to the object. Other factors that can play a role include atmospheric drag, the shape and density of the object, and the presence of other gravitational forces from celestial bodies.

5. How does the minimum speed for Earth orbit compare to escape velocity?

The minimum speed for Earth orbit is significantly lower than the escape velocity, which is the speed required for an object to break free from Earth's gravitational pull. The escape velocity for Earth is approximately 25,000 miles per hour, or 40,000 kilometers per hour, which is higher than the minimum orbital velocity. This means that an object with the minimum speed for Earth orbit can maintain a stable orbit, but it does not have enough speed to break free from Earth's gravity.

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