A city planner is working on the redesign of a hilly portion of the road. An important consideration is how steep the road can be so that even a low-powered car can get up the hill without slowing down. A particular small car with a mass of 1368 kg can accelerate on a flat level road from rest to 28.2 m/s in 16.3 seconds. Using this data, calculate the maximum steepness of the hill.
To calculate the maximum steepness of the hill, we need to determine the maximum force that the car's engine can produce to overcome the gravitational force pulling it downhill. This maximum force is equal to the net force acting on the car while it accelerates on a flat level road.
First, let's calculate the net force acting on the car. We can use Newton's second law, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
F = m * a
The mass of the car is given as 1368 kg, and the acceleration is calculated by dividing the change in velocity (28.2 m/s - 0 m/s) by the time it takes to achieve that change (16.3 seconds):
a = (28.2 m/s - 0 m/s) / 16.3 s
Now we can calculate the net force:
F = 1368 kg * [(28.2 m/s - 0 m/s) / 16.3 s]
Next, we need to determine the gravitational force acting on the car while it's going up the hill. The gravitational force (F_gravity) is given by the equation:
F_gravity = mass (m) * gravitational acceleration (g)
where the gravitational acceleration is approximately 9.8 m/s².
F_gravity = 1368 kg * 9.8 m/s²
Now that we have the net force and the gravitational force, the maximum steepness (angle θ) can be calculated using the equation:
sin(θ) = (net force - gravitational force) / net force
θ = arcsin((net force - gravitational force) / net force)
Calculating θ:
θ = arcsin((F - F_gravity) / F)
Now we can substitute the values into the equation and calculate the maximum steepness of the hill.
To calculate the maximum steepness of the hill, we need to determine the maximum angle at which the car can maintain its speed without slowing down. This can be achieved by analyzing the forces acting on the car on the hill.
First, let's calculate the acceleration of the car on the flat road using the given data. The initial velocity (u) is 0 m/s, the final velocity (v) is 28.2 m/s, and the time taken (t) is 16.3 seconds.
Using the formula: acceleration (a) = (v - u) / t
Substituting the values:
a = (28.2 m/s - 0 m/s) / 16.3 s
a ≈ 1.73 m/s²
Now, let's consider the forces acting on the car when it is on the hill:
1. Gravity Force (Fg): This force acts vertically downward with a magnitude of mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s²).
Fg = m * g
Fg = 1368 kg * 9.8 m/s²
Fg ≈ 13406.4 N
2. Normal Force (Fn): This force acts perpendicular to the hill and balances the vertical component of the gravitational force. Its magnitude is equal to the gravitational force.
Fn = Fg
Fn ≈ 13406.4 N
3. Friction Force (Ff): This force opposes the motion of the car and acts parallel to the hill. Its magnitude depends on the steepness of the hill and the coefficient of friction between the tires and the road surface.
On the maximum steepness, the force of friction will be at its maximum. This is given by the equation:
Ff = µ * Fn
where µ represents the coefficient of friction.
Since we want to find the maximum steepness of the hill, we need to find the minimum coefficient of friction that allows the car to maintain speed.
Now, let's set up the equation to find the maximum angle (θ) of the hill:
Ff = m * a
µ * Fn = m * a
µ * Fg = m * a
µ * 13406.4 N = 1368 kg * 1.73 m/s²
µ ≈ (1368 kg * 1.73 m/s²) / 13406.4 N
µ ≈ 0.177
Now that we have the coefficient of friction, we can find the maximum angle (θ) using the formula:
θ = arctan(µ)
θ = arctan(0.177)
θ ≈ 9.82°
Therefore, the maximum steepness of the hill is approximately 9.82°.