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#

Escape Velocity

Escape velocity is defined as the smallest speed that we need to give
an object in order to allow it to completely escape from the
gravitational pull of the planet on which it is sitting. To calculate
it we need only realize that as an object moves away from the center of
a planet, its kinetic energy gets converted into gravitational potential
energy. Thus we need only figure out how much gravitational potential energy an object gains as it moves from the surface of the planet off to infinity.
According to the above discussion for a planet with mass *M* and radius
*R*, this gain in gravitational potential energy is *GmM*/*R*.
For an object to just barely escape to infinity (without any residual
speed), all its initial kinetic energy must go into this increase in
gravitational potential energy. Thus, the initial kinetic energy must be
equal to *GmM*/*R*. Since kinetic energy is *mv*^{2}/2, equating these two
expressions tells us that the square of the initial velocity must
be equal to twice the gravitational potential energy divided the inertial
mass of the object. However, since gravitational potential energy is
proportional to inertial mass, we find finally that the square
of the escape velocity
depends only on the mass and radius of the planet (and of course Newton's
gravitational constant):
*v*^{2} =

Note that the inertial mass of the object
has cancelled, so that the escape velocity
of any object is independent of its mass. This means that if you want
to throw a grain of rice or an elephant into outer space, you need to
give them both the same initial velocity which for the Earch works
out to be about 10,000 meters per second.
You will also notice from the above expression that if the mass of a planet
or star stays fixed, but its radius decreases, then the escape velocity
necessarily increases. This will
play an important role in our discussion of black holes later in this
chapter.

** Next:** Orbital Motion and Kepler's
**Up:** Gravity
** Previous:** Gravitational Potential Energy
*modtech@theory.uwinnipeg.ca *

1999-09-29