Problem 239 - Calculus Volume 1

Question

For a given motor vehicle, the maximum achievable deceleration from braking is approximately 7 \(\mathrm{m} / \mathrm{sec}^{2}\) on dry concrete. On wet asphalt, it is approximately 2.5 \(\mathrm{m} / \mathrm{sec}^{2}\) Given that 1 \(\mathrm{mph}\) corresponds to \(0.447 \mathrm{m} / \mathrm{sec},\) find the total distance that a car travels in meters on dry concrete after the brakes are applied until it comes to a complete stop if the initial velocity is 67 \(\mathrm{mph}(30 \mathrm{m} / \mathrm{sec})\) or if the initial braking velocity is 56 \(\mathrm{mph}(25 \mathrm{m} / \mathrm{sec}) .\) Find the corresponding distances if the surface is slippery wet asphalt.

Step-by-Step Solution

Answer

On dry concrete, the car stops in 64.29 m from 67 mph and 44.64 m from 56 mph. On wet asphalt, it stops in 180 m from 67 mph and 125 m from 56 mph.

Step 1: Convert Velocities to Meters per Second

The initial velocities are already provided in meters per second: 67 mph is 30 m/s and 56 mph is 25 m/s. Since these values are already in the required units, we can proceed to the next step without conversion.

Step 2: Use the Formula for Stopping Distance

The stopping distance when decelerating is given by the formula: \( d = \frac{v^2}{2a} \), where \( v \) is the initial velocity in m/s, and \( a \) is the deceleration in m/s².

Step 3: Calculate Stopping Distance on Dry Concrete

For dry concrete, the deceleration \( a = 7 \) m/s².For \( v = 30 \) m/s:\[d = \frac{30^2}{2 \times 7} = \frac{900}{14} \approx 64.29 \text{ meters}\]For \( v = 25 \) m/s:\[d = \frac{25^2}{2 \times 7} = \frac{625}{14} \approx 44.64 \text{ meters}\]

Step 4: Calculate Stopping Distance on Wet Asphalt

For wet asphalt, the deceleration \( a = 2.5 \) m/s².For \( v = 30 \) m/s:\[d = \frac{30^2}{2 \times 2.5} = \frac{900}{5} = 180 \text{ meters}\]For \( v = 25 \) m/s:\[d = \frac{25^2}{2 \times 2.5} = \frac{625}{5} = 125 \text{ meters}\]

Key Concepts

Deceleration

Deceleration is an important concept when calculating stopping distances. It refers to the rate at which a vehicle slows down, which is the negative acceleration. The higher the rate of deceleration, the quicker a vehicle can stop. The formula for deceleration is simple; it is the change in velocity over time. However, when calculating stopping distance, deceleration is represented in the formula as \[d = \frac{v^2}{2a}\]Where:

• \(d\) is the stopping distance.
• \(v\) is the initial velocity in meters per second.
• \(a\) is the deceleration in meters per second squared.

This formula shows that deceleration is a key factor that determines how much distance is required for a vehicle to come to a full stop after braking. Higher deceleration values result in shorter stopping distances, which is especially useful for understanding the stopping capabilities on different surfaces.

Initial Velocity Conversion

In stopping distance calculations, having the velocity in the correct units is crucial. Generally, speeds are given in miles per hour (mph), but calculations often need them in meters per second (m/s). This conversion is necessary because the standard unit for velocity in physics is m/s.
The conversion factor is that 1 mph is approximately 0.447 m/s. So, to convert from mph to m/s, you multiply the speed in mph by 0.447. In the given exercise, the initial velocities were already converted:

• 67 mph equals approximately 30 m/s.
• 56 mph equals approximately 25 m/s.

Ensuring that the initial velocity is in m/s allows for accurate calculation of stopping distances using our deceleration formula. It is imperative in physics to ensure all measurements are in the correct units, particularly when dealing with various formulas.

Surface Conditions

Surface conditions dramatically affect a vehicle's ability to stop. Different surfaces offer different levels of friction that influence the deceleration rate. For instance, a dry concrete surface has a higher friction level than a wet asphalt surface, leading to greater deceleration values.
In the exercise, we looked at two different surface conditions:

• On dry concrete, a deceleration of \(7 \text{ m/s}^2\) was achievable.
• On wet asphalt, the maximum deceleration decreased to \(2.5 \text{ m/s}^2\).

This significant change in deceleration values shows how critical surface conditions are. Wet surfaces result in longer stopping distances because the lower friction reduces the rate of deceleration. Understanding how surface conditions impact stopping distances is crucial for safe driving, especially in adverse weather conditions.