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Non-conservative forces

It is important to note, however, that nonconservative forces — such as air resistance and rolling resistance — are a major factor in determining the speed of a tricycle going downhill. To account for these effects, we need to calculate generalized forces. Fortunately, since our problem only has a single independent variable — that of distance from the summit — calculating the generalized forces should be straightforward.

Rolling resistance is proportional to the normal force. The normal force could be calculated as:

The rolling resistance force is defined as:

Where $C_{rr}$ is the coefficient of rolling resistance. Substituting Equation (1) into Equation (1) yields:

This force directly opposes motion in the direction of $s$.

As for air resistance (or drag):

Substituting Equation (??) into Equation (3) yields:

Where $\rho$ is air density in $\mathrm{kg} \cdot \mathrm{m}^{-3}$, $C_D$ is a dimensionless drag coefficient, $A$ is the cross sectional surface area (in $m^2$) that the area acts on. This is also in the opposite direction to motion. Hence the generalized non-conservative forces acting on our tricycle would be: