Question

A car driver, driving at velocity \(v_{0}\), suddenly puts on the brake. What braking distance \(d\) is needed to stop the car? One can derive, from basic physics, that $$ d=\frac{1}{2} \frac{v_{0}^{2}}{\mu g} $$ Make a program for computing \(d\) in \((4.7)\) when the initial car velocity \(v_{0}\) and the friction coefficient \(\mu\) are given on the command line. Run the program for two cases: \(v_{0}=120\) and \(v_{0}=50 \mathrm{~km} / \mathrm{h}\), both with \(\mu=0.3\) \((\mu\) is dimensionless). (Remember to convert the velocity from \(\mathrm{km} / \mathrm{h}\) to \(\mathrm{m} / \mathrm{s}\) before inserting the value in the formula!) Name of program file: stopping_length.py.

Short answer

Braking distances are 189.57 m (120 km/h) and 32.02 m (50 km/h).

Step-by-step solution

Convert velocity units

Convert the velocity from kilometers per hour (km/h) to meters per second (m/s). Use the conversion factor: 1 km/h = 0.27778 m/s. Thus, \( v_{0_{m/s}} = v_{0_{km/h}} \times 0.27778 \).

Understand the formula for braking distance

The formula for braking distance is \( d = \frac{1}{2} \frac{v_0^2}{\mu g} \), where \( v_0 \) is the initial velocity in m/s, \( \mu \) is the friction coefficient, and \( g \) is the acceleration due to gravity, approximately 9.81 m/s².

Implement the formula in a Python program

Create a Python program file named `stopping_length.py`. In this program, define a function to calculate the distance using the formula. Use the converted velocity and given friction coefficient as inputs to the function.

Calculate for given cases

Run the program with two different initial velocities: \( v_0 = 120 \) km/h and \( v_0 = 50 \) km/h, both with \( \mu = 0.3 \). Convert these velocities to m/s: \( v_{0_1} = 120 \times 0.27778 = 33.33 \) m/s and \( v_{0_2} = 50 \times 0.27778 = 13.89 \) m/s. Substitute these into the distance formula.

Key concepts

Braking Distance Calculation

Calculating the braking distance is crucial for understanding how long it takes for a vehicle to come to a complete stop. The formula for calculating braking distance is derived from basic physics principles and is given by:

\[ d = \frac{1}{2} \frac{v_0^2}{\mu g} \]
where:

• \( d \) is the braking distance.
• \( v_0 \) is the initial velocity of the vehicle in meters per second (m/s).
• \( \mu \) is the friction coefficient between the car tires and the road, a dimensionless quantity.
• \( g \) is the acceleration due to gravity, which is approximately 9.81 m/s².

This formula shows that the braking distance is directly influenced by the square of the car's speed and inversely by the friction coefficient and gravity. A higher velocity or a lower friction coefficient results in a longer braking distance.

Velocity Conversion

In physics, maintaining consistent units is crucial for accurate calculations. As stated in the exercise, velocities given in kilometers per hour (km/h) need to be converted to meters per second (m/s) before being used in the braking distance formula.

The conversion is done using the formula:
\[ v_{0_{\text{m/s}}} = v_{0_{\text{km/h}}} \times 0.27778 \]
where:

• 1 km/h is equivalent to approximately 0.27778 m/s.
• The conversion factor ensures that arithmetic operations in the braking distance formula are coherent in the SI unit system.

Consider an initial velocity of 120 km/h: converting, we compute \( v_{0} = 120 \times 0.27778 \approx 33.33 \text{ m/s} \). Similarly, for 50 km/h, the conversion yields \( v_{0} = 13.89 \text{ m/s} \). This conversion is key to accurately applying the braking formula.

Friction Coefficient

The friction coefficient, represented by \( \mu \), plays a significant role in determining the braking distance. It is a dimensionless number that represents the resistance experienced when two surfaces slide against each other. For vehicles, this often pertains to the interaction between tires and the road surface.

A higher friction coefficient indicates more grip and results in a shorter stopping distance. Conversely, a lower coefficient means less grip, leading to longer stopping distances.

Several factors affect the value of \( \mu \):

• Road conditions: Dry roads offer higher friction compared to wet or icy surfaces.
• Tire quality: Newer tires often have better grip than worn-out ones.
• Material types: Different materials on tires or roads can significantly impact \( \mu \).

In our scenarios, \( \mu \) is given as 0.3, indicating relatively low friction, common in scenarios with adverse driving conditions.

Python Programming for Physics

Incorporating programming into physics calculations allows for efficient and accurate computations. Python, in particular, is a versatile tool for these applications.

For this exercise, you can create a simple Python script to compute the braking distance from the given formula:
1. Start by creating a file named `stopping_length.py`. 2. Define a function, perhaps `calculate_braking_distance`, which takes velocity and friction coefficient as arguments. 3. Within this function, implement the formula: - First, convert the velocity from km/h to m/s using the provided conversion factor. - Apply the braking distance formula. 4. Use the converted velocities for different cases and print out the computed braking distances.
Python's readability and simplicity make it ideal for physics-based calculations. Programming empowers students and professionals to handle complex concepts and solve them computationally.