The above equation clearly represents a conservation law, of some description, since the left-hand side only contains quantities evaluated at the initial height, whereas the right-hand side only contains quantities evaluated at the final height. In order to clarify the meaning of Eq. (123), let us define the

and the

Note that kinetic energy represents energy the mass possesses by virtue of its

(126) |

Incidentally, the expressions (124) and (125) for kinetic and gravitational potential energy,
respectively, are quite general, and do not just apply to free-fall under gravity.
The mks unit of energy is called the *joule* (symbol J). In fact, 1 joule
is equivalent to 1 kilogram meter-squared per second-squared, or 1 newton-meter. Note that
all forms of energy are measured in the *same* units (otherwise the idea of energy conservation
would make no sense).

One of the most important lessons which students learn during their studies
is that there are generally many different paths to the same
result in physics. Now, we have already analyzed free-fall under gravity using Newton's
laws of motion. However, it is illuminating to re-examine this problem from the
point of view of energy conservation. Suppose that a mass is dropped from rest and
falls a distance . What is the final velocity of the mass? Well,
according to Eq. (123), if energy is conserved then

(128) |

(129) |

Suppose that the same mass is thrown upwards with initial velocity . What is the
maximum height to which it rises? Well, it is clear from Eq. (125) that as
the mass rises its potential energy *increases*. It, therefore, follows from
energy conservation that its kinetic energy must *decrease* with height. Note, however,
from Eq. (124), that kinetic energy can never be negative (since it
is the product of the two positive definite quantities, and ). Hence, once the
mass has risen to a height which is such that its kinetic energy is reduced to *zero*
it can rise no further, and must, presumably, start to fall. The change in potential
energy of the mass in moving from its initial height to its maximum height is
. The corresponding change in kinetic energy is
; since
is the initial kinetic energy, and the final kinetic energy is zero.
It follows from Eq. (127) that
, which can be
rearranged to give

(130) |

It should be noted that the idea of energy conservation--although extremely useful--is
*not* a replacement for Newton's laws of motion. For instance, in the previous example, there
is no way in which we can deduce *how long* it takes the mass to rise to its
maximum height from energy conservation alone--this information can only come from the
direct application of Newton's laws.