When one deals with nonconservative forces, mechanical energy is "lost". This loss can be defined with the following
where Q is the mechanical energy lost, and K0 and
U0 refer to the initial kinetic
and potential energies.
Example:
Question:
If a car starts up a hill traveling at 25.0 m/s in neutral gear, what vertical distance up the hill will the car travel before coasting to a stop. Assume a frictional energy loss of 40%.
Answer:
A fractional energy loss of 40% means that Q = 0.4 (K0 + U0). Therefore:K + U = 0.6 (K0 + U0). When the car comes to a stop, the kinetic energy is zero, or K = 0. This means:
Note that without friction it would have gone 31.1 m, and that we don't need the slope of the hill unless we want to find how far it went along the slope.
U = mgh = 0.6 ( mv2 + 0)
\[ \rm \mathbf{\Rightarrow h = \frac{0.3 v^{2}}{g}} \] \[ \rm = \frac{0.3 \cdot(25 m/s)^{2}}{9.81 m/s^{2}} = 19.1 m \]