Additional Velocity Required for a Satellite Already in Orbit to Escape

[SeventeenForever]

Homework Statement

A satellite is orbiting the Earth around an orbit of radius R=2.5R₀, where R₀ is Earth's radius. What additional velocity is needs to be directed along the radius of the orbit so that satellite escapes Earth's gravity?

Homework Equations

Total Energy= K + U
Conservation of Energy: K₁ + U₁ = K₂ + U₂
K=0.5mv²
U=-Gm₁m₂ / R_{1, 2}

The Attempt at a Solution

Hi guys! We worked some similar problems in school and I've been trying to follow that solving this problem. We've always used conservation of mechanical energy to do this.

Our condition for when the satellite has just barely escaped Earth's gravitational field is:

T₂ = K₂ + U₂ = 0

U₂ is effectively zero because R_{1, 2} (the distance between objects 1 and 2) is so great. K₂ is zero because if the object has just barely escaped gravity, it wouldn't have any velocity. That all makes good sense to me!

The initial energy is a little more confusing to me. There is definitely some gravitational potential energy:

U=-Gm₁m₂ / R_{1, 2}

I'll let mass, m, be the mass of the satellite and M_earth be the mass of the Earth. R_{1, 2} in this case would be 2.5R₀.

U₁=-GmM_earth/2.5R₀

We've solved problems launching stuff on Earth into space, where the object initially has no kinetic energy. K₁ - U₁ would equal zero, so K₁ = U₁:

0.5mv² = GmM_earth/2.5R₀

The mass of the satellite would cancel from each side, and I can plug in the known constants and get an answer (2.2 x 10⁵ m/s is what I got).

But... I'm not sure if there's really no initial kinetic energy. The satellite is in space and is orbiting at some velocity, so wouldn't it have kinetic energy? If so, how would I go about finding it? Also, this seems "too simple." Is there something I'm overlooking?

 

[Orodruin]

The satellite is in space and is orbiting at some velocity, so wouldn't it have kinetic energy?

Indeed it will. You need to find this velocity by using mechanics. Hint: What is the force on the satellite and what force is required to keep it in a circular orbit?

 

[gneill]

But... I'm not sure if there's really no initial kinetic energy. The satellite is in space and is orbiting at some velocity, so wouldn't it have kinetic energy? If so, how would I go about finding it? Also, this seems "too simple." Is there something I'm overlooking?

Look up: "specific mechanical energy" (or, "specific orbital energy"). When the mass of the object in orbit is negligible with respect to that of the primary, it is often simpler to work with "specific" energies which is energy per unit mass (such as J/kg).

You should be able to find or derive an equation for the orbital velocity of a satellite in a circular orbit of a given radius (see the hint by Orodruin above). With that velocity and the given distance you should be in a position to calculate the specific mechanical energy of the orbit.

Remember to keep in mind that velocity is a vector quantity and that velocities combine accordingly.

 

[SeventeenForever]

I believe there's only one force on the satellite: gravity.

The gravity would provide a radial acceleration (towards Earth) that keeps the orbit circular since the velocity is tangential. The radial acceleration and velocity are related by:

a_rad=v²/R

The gravitational force is:

F_{1, 2}= Gm₁M_earth / R_{1, 2}²

The acceleration is F_{1, 2} divided by the mass of the satellite, m₁, making the acceleration:

GM_earth / R_{1, 2}² = v² / R_{1, 2}

GM_earth / R_{1, 2} = v²

I then take the square root of both sides to find the velocity. Is this the correct approach?

 

[gneill]

I then take the square root of both sides to find the velocity. Is this the correct approach?

Yup.