Problem 30 - College Physics

Question

(a) If the coefficient of kinetic friction between tires and dry pavement is \(0.80,\) what is the shortest distance in which you can stop an automobile by locking the brakes when traveling at \(29.1 \mathrm{~m} / \mathrm{s}\) (about \(65 \mathrm{mi} / \mathrm{h}) ?\) (b) On wet pavement, the coefficient of kinetic friction may be only \(0.25 .\) How fast should you drive on wet pavement in order to be able to stop in the same distance as in part (a)? (Note: Locking the brakes is not the safest way to stop.)

Step-by-Step Solution

Answer

(a) The shortest stopping distance on dry pavement is 54.1 m; (b) On wet pavement, you should drive at 16.3 m/s to stop in the same distance.

Step 1: Identify the Known Values

We are given the coefficient of kinetic friction for dry pavement \( \mu_k = 0.80 \) and for wet pavement \( \mu_k = 0.25 \). Additionally, the initial speed of the car is \( v = 29.1 \, \mathrm{m/s} \). We need to find the shortest stopping distance for part (a) and the speed for part (b) that results in the same stopping distance.

Step 2: Calculate Stopping Distance on Dry Pavement

When the brakes are locked, the only force stopping the car is the frictional force, given by \( F_{friction} = \mu_k \cdot m \cdot g \). Using the work-energy principle, the work done by friction is equal to the change in kinetic energy: \( \mu_k \cdot m \cdot g \cdot d = \frac{1}{2} m v^2 \). Simplifying, we find \( d = \frac{v^2}{2 \mu_k g} \), where \( g \approx 9.81 \, \text{m/s}^2 \). Substituting the values for dry pavement: \( d = \frac{(29.1)^2}{2 \times 0.80 \times 9.81} \approx 54.1 \, \text{m} \).

Step 3: Determine Speed on Wet Pavement for Same Stopping Distance

To stop in the same distance on wet pavement, the stopping distance formula \( d = \frac{v^2}{2 \mu_k g} \) should remain equal to 54.1 m. Re-arrange the formula to solve for initial speed on wet pavement: \( v = \sqrt{2 \mu_k g d} \). Substitute \( \mu_k = 0.25 \) for wet pavement and solve: \( v = \sqrt{2 \times 0.25 \times 9.81 \times 54.1} \approx 16.3 \, \text{m/s} \).

Key Concepts

Stopping Distance

Stopping distance is an essential concept to understand when dealing with vehicle safety and dynamics. It refers to the total distance a car travels before it comes to a complete stop after the brakes are applied. Several factors influence stopping distance, including the friction between the tires and the road, vehicle speed, and braking technique. Additionally, environmental conditions such as road surface, weather, and vehicle load play significant roles.

When discussing stopping distance, we consider two main components:

• Reaction Distance: The distance your car travels during your reaction time (the time it takes you to recognize a hazard and begin braking).
• Braking Distance: The distance from the point at which you start to apply the brakes until the vehicle stops.

In our exercise, the focus is on braking distance, particularly how it is affected by different coefficients of friction under dry and wet conditions. Understanding these components not only improves safety awareness but also aids in making informed decisions while driving.

Coefficient of Friction

The coefficient of friction is a crucial factor in determining how quickly a vehicle can stop. It's a measure of the "grip" between two surfaces, in this case, the car tires and the road. This measure is a dimensionless number typically ranging between 0 and 1.

There are two types of friction:

• Static Friction: Which is the frictional force preventing motion when the surfaces are at rest relative to each other.
• Kinetic Friction: Which occurs when the surfaces are sliding past each other. This is what we focus on when considering stopping a moving vehicle.

In our example, the change in the coefficient of kinetic friction from 0.80 on dry pavement to 0.25 on wet pavement indicates lesser grip in wet conditions. It's important to remember that a lower coefficient means less friction, leading to longer stopping distances. Therefore, understanding this concept can guide safer driving practices, especially under adverse weather conditions.

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that connects the work done by forces applied to an object and the change in its kinetic energy. In simpler terms, when work is done on an object, it either gains or loses energy.

For a decelerating vehicle, like in our exercise, the work done by the frictional force is what causes the vehicle to stop. The kinetic energy formula is expressed as:

• \[ \text{Kinetic Energy} = \frac{1}{2} m v^2 \]

In this context, the work done by friction to stop the car can be equated to the change in kinetic energy:

• \[ \mu_k \cdot m \cdot g \cdot d = \frac{1}{2} m v^2 \]

Solving for the stopping distance \(d\), we derive:

• \[ d = \frac{v^2}{2 \mu_k g} \]

This crucial equation relates stopping distance to initial speed, friction coefficient, and gravitational force—demonstrating how energy principles apply in real-world physics problems.

Physics Problem Solving

Physics problem solving involves breaking down complex scenarios into manageable parts, using scientific principles to find solutions. The exercise provided is a classic case, and here is a methodical approach to tackling such problems:

Step-by-Step Layout:

• Identify Known Values: Start by listing all given quantities and constants, such as coefficients of friction and initial speeds, which provide a foundation for problem-solving.
• Use Appropriately Linked Equations: Equations like those derived from the work-energy principle help connect different knowns and unknowns, facilitating comprehensive solutions.
• Re-arrange Formulas: Be comfortable manipulating equations to isolate the variable of interest, whether it's a stopping distance or initial speed.
• Substitute Values Cautiously: After structuring the relevant formula, substitute the known values to find the unknowns accurately.

Solving physics problems is akin to piecing a puzzle together, where understanding each small piece is crucial to find the whole picture. With practice, insights gained from exercises like this improve not only physics knowledge but also logical reasoning and analytical skills.