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Gravitational Escape Velocity Derivation
by Ron Kurtus (6 January 2010)
The equation for the gravitational escape velocity for an object moving freely directly upward, ignoring the effects of rotation and other gravitational fields, is ve = √(2GM/R).
This equation is obtained by applying the Law of Conservation of Energy to an object being projected upward against the downward gravitational force. The Law states that the total of the object's potential and kinetic energy is a constant. By comparing the potential and kinetic energy values at some given point with the values at infinity, you can determine the escape velocity equation.
Questions you may have include:
- What are the initial potential and kinetic energies of the object?
- What are the potential and kinetic energies at an infinite distance?
- What determines the final equation?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Initial PE and KE of object
Consider an object—such as a rocket—that is fired or projected upward until it reaches some height and velocity, at which time the engines are shut off.
At that position, the rocket would have an initial gravitational potential energy, and at that initial velocity, it would have an initial kinetic energy. The rocket would continue moving upward, slowing due to the gravitational force.
Initial potential energy
At the point where the engines shut off, the rocket has the potential of falling toward the ground. This is its initial gravitational potential energy:
PEi = GMm/Ri
where
- PEi is initial gravitational potential energy in kg-km2/s2
- G is the Universal Gravitational Constant = 6.67*10−20 km3/kg-s2
- M is the mass of the attracting object in kilograms (kg)
- m is the mass of the escaping object in kg
- Ri is the initial distance between the centers of mass of the objects in kilometers (km)
Note: Since escape velocity is usually stated in km/s, the units of PEi and R have been changed from stating them with meters. Likewise, the value to G has been changed to reflect use of kilometers.
Initial kinetic energy
An object projected upward—away from the Earth or other astronomical body—has kinetic energy, according to its mass and velocity. The initial kinetic energy it has is:
KEi = mvi2/2
where
- KEi is the initial kinetic energy in kg-km2/s2
- m is the mass of the object in kg
- vi is the initial upward velocity in km/s
Illustration of factors
The following picture shows the relationship of factors involved.
Factors involved in gravitational escape velocity
Total initial energy
Since the kinetic energy is moving upward and the potential energy is acting downward, the total energy at the initial position is:
Ti = KEi − PEi
Ti = mvi2/2 − GMm/Ri
Final PE and KE
Gravitational fields extend to infinity. Thus, if the initial velocity is great enough, the object will travel to an infinite distance.
Potential energy at infinity
The object's potential energy at an infinite distance is:
PE∞ = GMm/R∞
where
- PE∞ is the gravitational potential energy at infinity
- R∞ is the infinite distance between the objects
Since R∞ = ∞ (infinity), then PE∞ = 0.
Kinetic energy at infinity
The object's kinetic energy at an infinite distance is:
KE∞ = mv∞2/2
where
- KE∞ is the final kinetic energy
- v∞ is the final velocity
At infinity, the velocity of the object is zero: v∞ = 0. Thus KE∞ = 0.
Total final energy
Since the kinetic energy is moving upward and the potential energy is acting downward, the total energy at the initial position is:
T∞ = KE∞ − PE∞
T∞ = 0
Escape velocity equation
The Law of Conservation of Energy states that the total energy of a closed system remains constant. In this case, the closed system consists of the two objects, and no outside energy is affecting either object.
Thus the total final energy—potential energy plus kinetic energy—must equal the total initial energy:
Ti = T∞
KEi − PEi = 0
KEi = PEi
mvi2/2 = GMm/Ri
vi2 = 2GM/Ri
Since the initial velocity is sufficient for escape, vi = ve and:
ve = √(2GM/Ri)
where ve is the initial velocity required to escape the gravitational force of an astronomical object of mass M, from a distance of Ri from the center of the object.
Summary
The equation for the gravitational escape velocity is ve = √(2GM/R). By applying the Law of Conservation of Energy to an object being projected upward against the downward gravitational force, you can compare the potential and kinetic energy values at some given point with the values at infinity and determine the escape velocity equation.
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Resources
The following resources can be used for further study on the subject.
Web sites
What is escape velocity? - From PhysLink
Escape Velocity - From Wikipedia
Books
Top-rated
books on Escape Velocity and Space Travel
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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