Power Needs of a Car to Climb a Hill
a Consider a 1200-kg car cruising steadily on a level road at 90 km/h. Now the car starts climbing a hill that is sloped 30° from the horizontal (Fig. 2-37). If the velocity of the car is to remain constant during climbing, determine the additional power that must be delivered by the engine.
A car is to climb a hill while maintaining a constant velocity. The additional power needed is to be determined.
Analysis The additional power required is simply the work that needs to be done per unit time to raise the elevation of the car, which is equal to the change in the potential energy of the car per unit time:
\begin{aligned}\dot{W}_{g} &=m g \Delta z / \Delta t=m g V_{\text {vertical }} \\&=(1200 \mathrm{~kg})\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)(90 \mathrm{~km} / \mathrm{h})\left(\sin 30^{\circ}\right)\left(\frac{1 \mathrm{~m} / \mathrm{s}}{3.6 \mathrm{~km} / \mathrm{h}}\right)\left(\frac{1 \mathrm{~kJ} / \mathrm{kg}}{1000 \mathrm{~m}^{2} / \mathrm{s}^{2}}\right) \\&=147 \mathrm{~kJ} / \mathrm{s}=147 \mathrm{~kW} \quad \text { (or } 197 \mathrm{hp} \text { ) }\end{aligned}
Discussion Note that the car engine will have to produce almost 200 \mathrm{hp} of additional power while climbing the hill if the car is to maintain its velocity.