Escape Speed of a Satellite

[brometheus]

A satellite of mass 7500 kg orbits the Earth in a circular orbit of radius of 7.3(10^6) m (this is above the Earth's atmosphere). The mass of the Earth is 6.0(10^24) kg.

What is the minimum amount of energy required to move the satellite from this orbit to a location very far away from the Earth?


We are supposed to employ the Energy Principle to solve this problem, so we start with:

K_i + U_i = K_f + U_f

We know that K (at low speeds) = (1/2)*m*(v^2) and U = -GmM/r


Using the Energy Principle, we know that

K_f - K_i = U_i - U_f

ΔK = -ΔU = -GmM[(1/r_f) - (1/r_i)]

Since r_f is very large, ΔK = GmM(1/r_i)

Using accepted and aforementioned values,
ΔK = [6.7(10^-11) * 7500 * 6(10^24)]/[7.3(10^6)]

This got me approximately 4.13(10^11)J, which is apparently incorrect. What am I doing wrong?

 

[Andrew Mason]

A satellite of mass 7500 kg orbits the Earth in a circular orbit of radius of 7.3(10^6) m (this is above the Earth's atmosphere). The mass of the Earth is 6.0(10^24) kg.

What is the minimum amount of energy required to move the satellite from this orbit to a location very far away from the Earth?We are supposed to employ the Energy Principle to solve this problem, so we start with:

K_i + U_i = K_f + U_f

We know that K (at low speeds) = (1/2)*m*(v^2) and U = -GmM/rUsing the Energy Principle, we know that

K_f - K_i = U_i - U_f

ΔK = -ΔU = -GmM[(1/r_f) - (1/r_i)]

Since r_f is very large, ΔK = GmM(1/r_i)

Using accepted and aforementioned values,
ΔK = [6.7(10^-11) * 7500 * 6(10^24)]/[7.3(10^6)]

This got me approximately 4.13(10^11)J, which is apparently incorrect. What am I doing wrong?

You are forgetting that it already has kinetic energy. If it gained GmM/R in kinetic energy it would have more than enough energy to escape.

In order to escape, it has to have just enough kinetic energy to get to an arbitrarily large distance from the Earth ie. where it has 0 kinetic and 0 potential energy.

So the condition for escape is: KE_∞ + U_∞ = 0

Can you write the equation for KE_r+ U_r (hint: if there are no external forces acting, does KE + U change?)

Welcome to PF by the way!

AM

 

[Alioth]

the binding energy of a satellite moving in an orbit is GMm/2r. so i think you ought to divide your answer by 2. [4.13*(10^11)] will be the answer if it is at rest on the Earth's surface.
PS: i am new here. could you please tell me how to start a new thread?

 

[brometheus]

Ah, okay, that makes sense now. Thanks for your help, and the warm welcome!

To start a new thread I just went to the specific forum (Introductory Physics in this case) and clicked "New Topic" at the top right. It's in the same location as "New Reply" on this page.

 

[Nickie]

It seems like you have made a slight error in your calculation. The correct value for the change in kinetic energy should be 4.13(10^7)J, not 4.13(10^11)J. This is because the units for the mass of the satellite should be in kilograms (kg), not grams (g). So the correct calculation would be:

ΔK = [6.7(10^-11) * 7500 * 6(10^24)]/[7.3(10^6)] = 4.13(10^7)J

This means that the minimum amount of energy required to move the satellite from its current orbit to a location very far away from the Earth is 4.13(10^7)J. This energy is known as the escape energy or escape velocity.

To put this into perspective, the escape velocity for the Earth is approximately 11.2 km/s. This means that the satellite would need to be accelerated to a speed of at least 11.2 km/s to escape the Earth's gravitational pull and travel to a location far away from the Earth.

It is important to note that this calculation assumes there is no external force acting on the satellite, such as the thrust from a rocket. In reality, a satellite would need to overcome not only the Earth's gravitational pull, but also the drag and atmospheric resistance from the Earth's atmosphere.