We considered the phrase What goes up must come down. We decided to look at this question from an energy perspective. If an object is going up (from the surface of the Earth), it is moving and thus has kinetic energy. The acceleration due to gravity is opposite to the direction of the velocity and thus the object is slowing down -- or in energy language the kinetic energy is being converted to gravitational potential energy. But the question is whether must we always reach a point where the object turns around -- where all of its kinetic energy is converted to potential energy.

Since we are asking the question about whether we can get far from the surface of the Earth, we cannot use the simplified gravitation potential energy m g h for that applies only near the surface. Instead we must the more general form U_grav = - G M m / R. Note this formula looks similiar to Newton's form for the gravitational force F = - G M m / R², but has only one R in the denominator while the force has R². (They are of course similar looking because the energy is related to the integral of the force.)

E_total = m v² / 2 - G M m / R

where 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. We say that we can "escape" the Earth if R goes to infinity. If we take R to infinity and v to zero, we have the situation of just being able to escape, and if we plug these values in above we get E_total = 0. Because energy is conserved, if the energy is zero or higher at the surface on the Earth, it remains so -- unless something from the outside the system (not the Earth) does work on the object.

m v_escape² / 2 - G M m / R = 0

Solving for v yields the so-called escape velocity.

v_escape = ( 2 G M / R )^1/2.

If one has this velocity or more when one is at the Earth's surface then one can escape the Earth's gravitional field. We went on to say that for objects other than the Earth that the combination ( 2 G M / R )^1/2 might exceed the speed of light c. In such an instance then an object cannot escape since it cannot have a velocity greater than the speed of light. Considerations like these give rise to the concept of a Black Hole.

We also furthered our discussion of entropy and the Second Law of Thermodynamics -- trying to come to terms with what it means as well as what it allows and what it rules out. One of the earliest formulations of the law is that heat will flow spontaneously from a "place" of higher temperature to lower but not vice versa. We associated friction (which is a force) with the generation of heat (which is a less useful form of energy) and noted that friction is directed opposite to displacement -- it always "works against us."

Not done yet?