Understanding the Derivation of Escape Velocity
The Minimum Speed Needed to Escape a Gravitational Field
Mar 15, 2009
When NASA launches a rocket or satellite into space, it must move at a certain minimum speed to break the bonds of Earth's gravity. A rocket not reaching escape velocity falls back to Earth.
This concept applies to anything not just rockets. For example, a good baseball player might hit the ball out of the park. The ball does however fall back to Earth because its speed is less than the escape velocity of 11 kilometers/second. Now if superman were to start playing baseball and hit the ball at a speed of 11 kilometers/second (or more), then the ball would escape Earth's gravity and fly into orbit.
Physicists derive the concept of escape velocity from the principle of conservation of energy.
Conservation of Energy
The types of mechanical energy are kinetic energy and potential energy. Kinetic energy is the energy of motion, and potential energy is the energy of the position.
As an object moves upward its gravitational potential energy increases; as it falls downward gravitational potential energy decreases. In addition, as the object moves up, its speed, and therefore kinetic energy, decreases. As it falls back down its speed and kinetic energy increase.
Notice that as one type of energy increases the other type decreases. In the absence of friction, the total sum of the potential and kinetic energy must remain a constant. In physicists' language: the energy must be conserved.
When NASA launches a rocket into space, the fast moving rocket has a large kinetic energy. Once the motors stop firing, the rocket gets no additional energy. For the rocket to continue moving upward, its kinetic energy must be converted into potential energy.
Escape Velocity and Energy Conservation
To derive the formula for escape velocity from the conservation of energy law, physicists take the potential energy at an infinite distance as 0. The potential energy at the surface is then negative. To just barely escape the gravitational field, the kinetic energy at an infinite distance would also be 0. Hence the sum of the kinetic and potential energy equals 0 at the surface or any other point.
The kinetic energy (KE) is given by the formula:
KE=(1/2)mv^2.
The potential energy (PE) is given by the formula:
PE=-(GMm)/r.
Here m is the mass of the escaping object, M is Earth's mass, r is the distance between the escaping object and the center of Earth, G is the universal gravitational constant, and v is the speed of the escaping object. The more familiar form of the potential energy (PE=mgh) cannot be used here because this formula applies only near Earth's surface.
Setting the sum of the kinetic and potential energies at Earth's surface equal to zero and solving for the velocity gives the formula for the escape velocity:
v(escape)= square root(2GM/R)
Where R is Earth's radius. Putting the mass and radius of the Earth into this formula gives 11 kilometers/second for the escape velocity from Earth.
This formula can also be applied to any planet, moon, star, or other astronomical object by using the mass and radius of that particular object. For example the escape velocity from the Moon is 2.4 kilometers/second from this formula. Also, equating the escape velocity to the speed of light and using the mass of the Sun gives about 3 kilometers for the radius of the event horizon for a solar mass black hole.
Escape velocity is an important consequence of conservation of energy. It also provides a nice illustration of how fundamental physical principles have far reaching consequences.
Further Reading
Wilson, J.D., Buffa, A.J., and Lou, B., College Physics 6th ed., Pearson, 2007.
 |
