A city planner is working on the redesign of a hilly portion of a city. An important consideration is how steep the roads can be so that even low-powered cars can get up the hills without slowing down. It is given that a particular small car, with a mass of 1090 kg, can accelerate on a level road from rest to 18 m/s (64.8 km/h) in 13.0 s. Using this data, calculate the maximum steepness of a hill.

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°.

To calculate the maximum steepness of a hill, we need to use the concept of forces and Newton's laws of motion.

First, let's calculate the acceleration of the car on a level road. We know the initial velocity (0 m/s), the final velocity (18 m/s), and the time taken (13.0 s).

Using the formula:
acceleration = (final velocity - initial velocity) / time taken

acceleration = (18 m/s - 0 m/s) / 13.0 s
acceleration = 1.38 m/s^2

Now, let's calculate the force of gravity acting on the car. The force of gravity can be calculated using the formula:
force_gravity = mass * acceleration_due_to_gravity

where the acceleration due to gravity is approximately 9.8 m/s^2.

force_gravity = 1090 kg * 9.8 m/s^2
force_gravity = 10,682 N

Next, let's calculate the maximum force the car can handle on a hill without slowing down. This force is equal to the force of gravity acting on the car.

maximum_force = force_gravity
maximum_force = 10,682 N

Now, let's calculate the maximum steepness of a hill. The steepness of a hill can be determined using the concept of the sine function.

The equation for calculating the steepness of a hill is:
steepness = (maximum_force / mass) * sin(theta)

where theta is the angle of the hill.

Rearranging the equation to solve for the angle theta, we have:
theta = arcsin((maximum_force / mass) * 1/9.8)

Substituting the known values into the equation, we can calculate the maximum steepness of the hill.

theta = arcsin((10,682 N / 1090 kg) * 1/9.8)
theta = arcsin(0.9807)
theta ≈ 79.2 degrees

Therefore, the maximum steepness of the hill is approximately 79.2 degrees.