A 1600-kg car experiences a combined force of air resistance and friction that has the same magnitude whether the car goes up or down a hill at 32 m/s. Going up a hill, the car's engine produces 54 hp more power to sustain the constant velocity than it does going down the same hill. At what angle is the hill inclined above the horizontal?

54 hp = 40,284 Watts

Difference between uphill and downhill power = 2*(weight)*(speed)*sinA

sinA = 40,284/[2*1600*9.8*32] = 0.04014
A = 2.3 degrees

To find the angle of the hill inclined above the horizontal, we need to analyze the forces acting on the car while going up and down the hill.

Let's break it down step by step:

Step 1: Calculate the forces acting on the car.

The forces acting on the car can be divided into two components: the force of gravity and the combined force of air resistance and friction.

The force of gravity can be calculated using the formula:

F_gravity = m * g

where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s²).

Given that the mass of the car is 1600 kg, the force of gravity is:

F_gravity = 1600 kg * 9.8 m/s²

Step 2: Determine the net force acting on the car.

The net force acting on the car is the difference between the forces produced by the engine and the forces of air resistance and friction.

When going up the hill, the car's engine produces an additional 54 horsepower (hp) of power compared to going down the hill. To convert horsepower to watts (W), the unit for power in the International System of Units (SI), we use the conversion factor:

1 hp = 745.7 W

So, the additional power produced by the engine going up the hill is:

P_additional = 54 hp * 745.7 W/hp

Step 3: Calculate the magnitude of the combined force of air resistance and friction.

Since the combined force of air resistance and friction has the same magnitude whether going up or down the hill, we can express it as:

F_air_friction_up = F_air_friction_down = F_air_friction

Step 4: Calculate the net force going up the hill.

The net force going up the hill is given by:

F_net_up = F_gravity + F_air_friction + P_additional

Step 5: Calculate the net force going down the hill.

The net force going down the hill is given by:

F_net_down = F_gravity + F_air_friction

Step 6: Set up equations for the net forces.

Using the above calculations, we can set up two equations for the net forces:

F_net_up = F_gravity + F_air_friction + P_additional
F_net_down = F_gravity + F_air_friction

Step 7: Solve for the unknowns.

We have two equations with two unknowns: F_air_friction and the angle of the hill.

Since the magnitude of the combined force of air resistance and friction is the same going up and down the hill, we can equate the two net forces:

F_gravity + F_air_friction + P_additional = F_gravity + F_air_friction

By canceling out the forces of gravity on both sides and rearranging the equation, we can isolate P_additional:

P_additional = F_air_friction

Step 8: Substitute back into the equation.

Substituting P_additional = F_air_friction into the equation for F_net_up, we have:

F_net_up = F_gravity + P_additional + P_additional

Step 9: Solve for the angle of the hill.

Using the formula for the net force on an inclined plane:

F_net_up = m * g * sin(theta)

where theta is the angle of the hill, we can rearrange the equation to solve for theta:

sin(theta) = (F_net_up - F_gravity) / (m * g)

Now, we can plug in the known values to calculate the angle of the hill.