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Power conversion is the fundamental step in understanding how much work a machine, such as a car, can do based on its power source. It involves converting units of horsepower (hp) into watts (W), which are more commonly used in scientific calculations. Knowing that 1 horsepower is equivalent to 746 watts is essential. For example, if a car engine is using 8 hp, this translates to \(8 \times 746\) W, or 5968 W. This conversion is crucial because it provides us with a common unit that can then be used in further calculations such as force and energy needs, especially when performing tasks like going uphill or fighting drag and friction on a flat surface.

Retarding forces are the forces that work against the motion of the car, slowing it down. These include friction from the road and resistance from the air. To calculate the total retarding force when a car is cruising at a constant speed, we can use the formula \( P = F \times v \). Here, \( P \) is the power in watts, \( F \) is the force in newtons, and \( v \) is the speed in meters per second (converted from kilometers per hour). For our exercise, with a known power of 5968 W and a speed of 16.67 m/s, the retarding force comes out to about 357.8 N. This force needs to be constantly overcome by the engine to maintain a steady speed on a level road.

Driving uphill and downhill introduces additional forces into the power calculation. When driving uphill at a 10% grade, the car has to work against gravity. Calculating this involves recognizing that the hill’s rise adds an additional force component \( F_{g} = m \cdot g \cdot \sin(\theta) \). For a 10% grade, \( \sin(\theta) \approx 0.1 \), leading to an uphill gravitational force of 1764 N. Combined with the retarding force, the total force becomes 2121.8 N. The power needed is then calculated as \( P = F_{\text{total}} \times v \), which results in a significant increase in power requirements to about 47.4 hp.
On the other hand, when driving downhill, gravity aids in propulsion. This reduces the net retarding force, as calculated for a 1% grade, resulting in lower necessary power, around 4.06 hp.

Coasting occurs when the car moves at a steady speed without additional power from the engine. This happens when the gravitational force matches the retarding forces perfectly, requiring no extra energy from the engine. To find the downhill slope where coasting occurs, set the retarding force equal to the gravitational force component. The equation \( F_{g} = 357.8 \) N is used to solve for \( \sin(\theta) \), leading to approximately 0.0203. Converting this to a percentage gives a downhill grade of about 2.03%. This grade indicates a balance between gravity and the retarding forces, allowing the car to maintain speed without engine power.