To calculate the maximum steepness of a hill, we need to consider the forces acting on the car. One of the key forces that affects the car's ability to climb a hill is the gravitational force. As the car goes uphill, the gravitational force pulls it back down, opposing its motion.
We can start by calculating the force required for the car to accelerate from rest to 18 m/s in 13.0 s on the flat ground. This force is given by Newton's second law of motion, F = ma, where F is the force, m is the mass of the car, and a is the acceleration.
Given:
Mass of the car (m) = 1090 kg
Acceleration (a) = (18 m/s - 0 m/s) / 13 s = 1.385 m/s^2
Using F = ma, we can calculate the force:
F = (1090 kg) * (1.385 m/s^2) = 1508.65 N
Now, let's calculate the maximum steepness of a hill. On a hill, the force required to climb against gravity is given by F = mgsinθ, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and θ is the angle of the slope.
To find the maximum angle of the slope, we need to set the force required to climb against gravity (mgsinθ) equal to the force calculated earlier (1508.65 N):
mgsinθ = 1508.65 N
Rearranging the equation to solve for θ:
sinθ = 1508.65 N / (m*g)
θ = arcsin(1508.65 N / (m*g))
Substituting the known values:
θ = arcsin(1508.65 N / (1090 kg * 9.8 m/s^2))
Using a calculator, we can evaluate the angle:
θ ≈ arcsin(0.1414)
θ ≈ 8.06°
Therefore, the maximum steepness of the hill that the car can climb without slowing down is approximately 8.06°.