Work required to lift object from earth's surface to circular orbit

[aeromat]

Homework Statement

How much work must be done by a rocket engine to lift a 1000kg payload from rest on the Earth's surface to a circular orbit of radius 2Re (2 times the radius of the earth)?

Homework Equations

Eg = -GmM/r
Ek = GmM/2r

The Attempt at a Solution

I know ΔE = W
I know that the energies are the following:
Initial E {0 + Eg} <-- assuming its at rest
Final E {0 + Eg} <-- when it has reached this distance.

However, I don't know how I am going to solve this out, or whether or not what I did was correct...

 

[ideasrule]

Initial E {0 + Eg} <-- assuming its at rest

This is correct.

Final E {0 + Eg} <-- when it has reached this distance.

This is not. Kinetic energy can't be zero, or else the satellite would plummet to the ground instead of orbiting.

Once you figure out the final and initial energies, work done is just the difference between the two, as you've noted in your first equation.

 

[HallsofIvy]

Yes. Note that "Work required to lift object from Earth's surface to circular orbit" and "How much work must be done by a rocket engine to lift a 1000kg payload from rest on the Earth's surface to a circular orbit of radius 2Re (2 times the radius of the earth)?" are completely different questions!

 

[gneill]

It might be pertinent to note that "at rest on the Earth's surface" does not mean it truly motionless. The Earth rotates on its axis, so even while the rocket is "at rest" on the surface it can take advantage of an initial velocity (kinetic energy).

 

[rwisz]

I can provide a response to this question by using the concept of work and energy. The work required to lift an object from the Earth's surface to a circular orbit can be calculated by using the equation W = ΔE, where W is the work done and ΔE is the change in energy.

In this case, the initial energy of the object on the Earth's surface is purely gravitational potential energy, given by the equation Eg = -GmM/r, where G is the gravitational constant, m is the mass of the object, M is the mass of the Earth, and r is the distance from the object to the center of the Earth. The final energy of the object in a circular orbit is a combination of gravitational potential energy and kinetic energy, given by the equation Ek = GmM/2r.

To find the work required, we need to calculate the change in energy, which is equal to the final energy minus the initial energy. Therefore, ΔE = Ek - Eg = [GmM/2r] - [-GmM/r] = [3GmM/2r].

Substituting the values given in the problem, we get ΔE = [3(6.67x10^-11)(1000)(5.97x10^24)/2(2x6.37x10^6)] = 1.16x10^11 J.

Therefore, the work required to lift a 1000kg payload from rest on the Earth's surface to a circular orbit of radius 2Re is approximately 1.16x10^11 J.