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Solving physics problems often involves breaking down a scenario into known quantities and equations. In this case, we are dealing with a car moving up and down an inclined plane. The main task is to find the angle of inclination of the hill. It's important to clearly identify the given parameters, such as the car's mass and velocity, as well as any additional power required to maintain motion. In the problem, the car's mass is given as 1900 kg, and it moves at a constant speed of 27 m/s. The key detail is that the car needs an extra 47 horsepower to drive up the hill compared to coming down. Understanding this requirement helps us frame the problem in terms of physics concepts like force, work, and power. When solving such problems, always start by listing known values and what you need to find, as this creates a clearer path to the solution.

Gravitational force plays a significant role in this physics problem. It is the force that the engine must overcome to move the car upwards along an inclined plane. The gravitational force acting on the car can be calculated using the equation: \[ F_g = mg, \]where

• \( m \) is the mass of the object (in kilograms),
• \( g \) is the acceleration due to gravity, generally approximated as 9.8 m/s².

However, because the car is moving up an incline, only a component of this gravitational force acts along the plane. This component can be calculated as: \[ F = mg \sin \theta, \]where \( \theta \) is the angle of the incline. In this case, solving for \( \theta \) involves equating the component of gravitational force with the additional power required to move up the hill, as converted from horsepower to watts. Understanding this interaction between forces is vital in solving inclined plane problems.

Power conversion is an essential step in handling physics problems involving moving objects on inclined planes. The problem states that the car’s engine requires an additional 47 horsepower when climbing. To use this information in your calculations, you need to convert horsepower into watts, the standard unit of power in physics:\[ 1 \text{ hp} = 746 \text{ W} \Rightarrow 47 \text{ hp} = 47 \times 746 = 35062 \text{ W} \]This conversion is crucial since power sets the stage for our calculations and reveals how much work per unit time the engine does to overcome gravitational force. Always remember to convert all units appropriately to maintain consistency in your equations. By understanding and converting power correctly, you align with physics standards, making it easier to solve problems efficiently.

Trigonometry allows us to connect angles to side lengths in triangles, which is particularly useful in problems involving inclined planes. In this exercise, trigonometry helps us relate the angle of incline (\( \theta \)) to components of force. When a car moves uphill, only a portion of the gravitational force acts along the incline, directly calculated as \( mg \sin \theta \). This component equilibrates with the additional power needed (in watts). Once we set up this equation,\[ P = mg \sin \theta \times v, \]we can isolate \( \theta \) by solving for \( \sin \theta \) and use the inverse sine function to find the angle in degrees. As shown, \[ \sin \theta = \frac{P}{m \cdot g \cdot v}, \]where we've already calculated all terms, yielding \( \sin \theta \approx 0.0699 \). Using a calculator, \( \theta = \arcsin(0.0699) \approx 4.01^\circ \). Trigonometry thus bridges numerical values with physical dimensions, providing precise insights into naturally occurring phenomena.