Max Hill Steepness for Low-Powered Cars

[physicsss]

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 1500 kg, can accelerate on a level road from rest to 25m/s in 10.0 s. Using these data, calculate the maximum steepness of a hill.

 

[Tide]

What have you done so far on the problem?

 

[M98Ranger]

The maximum steepness of a hill that a low-powered car can handle is an important factor to consider in city planning, especially in hilly areas. This is because steep roads can pose a challenge for cars with less power, potentially slowing down traffic and causing safety concerns.

In order to calculate the maximum steepness of a hill for low-powered cars, we can use the given data of a small car with a mass of 1500 kg, accelerating from rest to 25m/s in 10.0 s on a level road. This information allows us to calculate the car's acceleration, which is 2.5 m/s^2 (25m/s divided by 10.0 s).

Now, let's assume that the car is traveling up a hill with a certain angle of incline, which we will call theta (θ). The force acting on the car in the direction of motion is the component of the car's weight parallel to the incline, which is given by Wsinθ, where W is the weight of the car.

We also know that the net force acting on the car is equal to its mass (m) multiplied by its acceleration (a). Therefore, we can set up the following equation:

Wsinθ = ma

Solving for the angle of incline (θ), we get:

θ = sin^-1 (ma/W)

Plugging in the values of m = 1500 kg, a = 2.5 m/s^2, and W = mg (where g is the acceleration due to gravity, which is 9.8 m/s^2), we get:

θ = sin^-1 [(1500 kg)(2.5 m/s^2)/(1500 kg)(9.8 m/s^2)]

θ = sin^-1 (0.255)

θ = 14.7°

Therefore, the maximum steepness of a hill that a low-powered car, with a mass of 1500 kg and an acceleration of 2.5 m/s^2, can handle is approximately 14.7°. This means that the incline of the hill should not exceed 14.7° in order for the car to maintain its speed without slowing down.

In conclusion, when redesigning hilly portions of a city, it is important to consider the maximum steepness of the roads for low-powered cars. By using the given data and the equation for calculating the angle of incline,