Fundamentals of Transportation/Grade/Solution

 Solution: The winner is obvious the Tour-de-France champion, as everything between the two is equal with the exception of power. Since the champion produces more power, he, by default, would win.

We need to find the maximum steady state speed for each racer to compute the differences in arrival time. At the steady state speed we must have:

$$ F_t  = R_a  + R_{rl}  + R_g  \,\!$$

Where:
 * $$F_t\,\!$$ = Tractive Effort
 * $$m\,\!$$ = Vehicle Mass
 * $$a\,\!$$ = Acceleration
 * $$R_a\,\!$$ = Aerodynamic Resistance
 * $$R_{rl}\,\!$$ = Rolling Resistance
 * $$R_g\,\!$$ = Grade Resistance

We also know:
 * $$\rho\,\!$$ = 1.0567 kg/cubic-m
 * $$W\,\!$$ = 756 N
 * $$f_{rl}\,\!$$ = 0.01
 * $$G\,\!$$ = 0.06
 * $$A\,\!$$ = 0.4 $$m^2$$
 * $$C_D\,\!$$ = 0.9

To estimate available tractive effort we can use the definition of power as time rate of work, P = FV, to get:

$$F = \frac\,\!$$

Substituting in the components of the individual formulas to the general one, we get:

$$\frac = \frac{\rho }{2}AC_D V ^2 + f_{rl}W + WG\,\!$$

If we compute rolling and grade resistance, we get:

$$R_{rl} = f_{rl}W = 0.01(756) = 7.56\ N\,\!$$

$$R_g = WG = 756(0.06) = 45.36\ N\,\!$$

Together, they add up to 52.92 N.

Aerodynamic Resistance can be found to be:

$$R_a = \frac{1.0567}{2}(0.4)(0.9)v ^2 = 0.19v ^2\,\!$$

With everything substituted into the general formula, the end result is the following formulation:

$$v = \frac\,\!$$

This problem can be solved iteratively (setting a default value for v and then computing through iterations) or graphically. Either way, when plugging in 510 watts for the Tour-de-France champion, the resulting velocity is 7.88 meters/second. Similarly, when plugging in 310 watts for the HYHM, the resulting velocity is 5.32 meters/second.

The hill is five miles in length, which translates to 8.123 kilometers, 8,123 meters. It will take the champion 1030 seconds (or 17.1 minutes) to complete this link, whereas the HYHM will take 1527 seconds (or 25.4 minutes). The resulting difference is 8.3 minutes.

/Solutions