A 1 420kg car coasts in neutral down a 2.0 degrees hill. The car attains a terminal velocity of 24.3 meters per second. How much power must the engine deliver to drive the car on a level road at 24.3 meters per second?
To find the power required to drive the car on a level road at the same velocity as it reaches while coasting down the hill, we need to consider the forces acting on the car.
First, let's calculate the gravitational force acting on the car while coasting down the hill. The gravitational force can be calculated using the formula:
Fg = m * g * sin(θ)
where:
Fg is the gravitational force,
m is the mass of the car (m = 1,420 kg),
g is the acceleration due to gravity (g = 9.8 m/s²),
θ is the angle of the hill (θ = 2.0 degrees).
Converting the angle from degrees to radians:
θ (in radians) = θ (in degrees) * (π / 180)
θ (in radians) = 2.0 * (π / 180) = 0.0349066 radians
Now we can calculate the gravitational force:
Fg = 1,420 kg * 9.8 m/s² * sin(0.0349066 radians)
Fg ≈ 49.83 N
At terminal velocity, the gravitational force is balanced by the drag force, which is given by:
Fd = 0.5 * ρ * A * Cd * v²
where:
Fd is the drag force,
ρ is the air density (ρ = 1.225 kg/m³),
A is the cross-sectional area of the car,
Cd is the drag coefficient of the car,
v is the velocity of the car (v = 24.3 m/s).
The cross-sectional area and drag coefficient will depend on the specific car model but we can assume some reasonable values. Let's say the cross-sectional area is 2.0 m² and the drag coefficient is 0.35.
Now we can calculate the drag force:
Fd = 0.5 * 1.225 kg/m³ * 2.0 m² * 0.35 * (24.3 m/s)²
Fd ≈ 644.475 N
Since the drag force at terminal velocity is equal to the gravitational force, we can deduce that the net force acting on the car on a level road at the same velocity must be zero.
Now, let's calculate the power required to overcome this net force on the level road. Power is given by:
P = F * v
where:
P is the power,
F is the net force on the car,
v is the velocity of the car (v = 24.3 m/s).
In this case, since the net force is zero, the power required will also be zero. This means that the engine doesn't need to deliver any power to maintain the car's velocity on a level road at 24.3 m/s, assuming there is no additional force like rolling resistance or other external factors.