A car of mass m has an engine that provides a constant power output P. Assuming no frictional losses, what is the maximum constant speed $v_{\max}$ that this car can drive up a long incline that makes an angle $θ$ with the horizontal?

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$v_{\max} = P / (m g \sin θ)$

$v_{\max} = P^{2} \sin θ / m g$

$v_{\max} = \sqrt{2 P / m g /} \sin θ$

There is no maximum constant speed.

answer is A.

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Detailed Solution

This problem instructs us to neglect friction. It is the friction with the air that we need to neglect, not the static friction with the ground, otherwise there would be no motion up the hill. With that ambiguity out of the way, let us look at the problem. If the car is to move up at a constant speed, the net force on it must be 0, so the force that propels the car, F, must be equal to the component of gravity parallel to the slope, $m g \sin θ .$ therefore,

$P = F v_{\max} = m g v_{\max} \sin θ \Rightarrow v^{\max} = \frac{P}{m g \sin θ}$