Whether you walk up a mountain or whether a car drives you up the mountain, the same amount of work
has to be done on you. (You weigh a certain number of newtons, and you have to be lifted up the same
distance either way!) But clearly there’s something different about walking up over the course of several
hours and driving up over several minutes. That difference is power.
Power is, thus, measured in units of joules/second, also known as watts. A car engine puts out hundreds of
horsepower, equivalent to maybe 100 kilowatts; whereas, you’d work hard just to put out a power of a
few hundred watts.
Potential Energy vs. Displacement Graphs
A different potential energy function can actually be derived for ANY conservative force. (A conservative
force means that energy is conserved when this force acts ... examples are gravity, spring, and
electromagnetic forces; friction and air resistance are the most common nonconservative forces.) The
potential energy U for a force is given by the following integral:
Note that this equation works for the gravitational force F = −mg (where − is the down direction) and the
spring force F = −kx ; the potential energy attributable to gravity integrates to mgh , and the spring
potential energy becomes ½kx 2 .
Chris on a Skateboard
Once a potential energy of an object is found as a function of position, making a U vs. x graph tells a lot
about the long-term motion of the object. Consider the potential energy of a spring, ½kx 2 . A graph of this
function looks like a parabola, as shown in Figure 14.2 .