5 CONSERVATION OF ENERGY 5.2 Energy conservation during free-fall
0
v^2 = v 2 − 2 g s. Suppose that the mass falls from height h 1 to h 2 , its initial velocity
is v 1 , and its final velocity is v 2. It follows that the net vertical displacement of
the mass is s = h 2 − h 1. Moreover, v 0 = v 1 and v = v 2. Hence, the previous
expression can be rearranged to give
1
m v 2 + m g h =1
m v 2 + m g h
2 1
1
2 2
2.^ (5.1)^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. (5.1), let us define the kinetic energy of the
mass,
K =1
m v^2 , (5.2)
2
and the gravitational potential energy of the mass,
U = m g h. (5.3)Note that kinetic energy represents energy the mass possesses by virtue of its
motion. Likewise, potential energy represents energy the mass possesses by virtue
of its position. It follows that Eq. (5.1) can be written
E = K + U = constant. (5.4)Here, E is the total energy of the mass: i.e., the sum of its kinetic and potential
energies. It is clear that E is a conserved quantity: i.e., although the kinetic and
potential energies of the mass vary as it falls, its total energy remains the same.
Incidentally, the expressions (5.2) and (5.3) for kinetic and gravitational po-tential 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 isthat there are generally many different paths to the same result in physics. Now,