Problem 92 - College Physics

Question

\(\bullet\) \(\bullet\) Automobile accident analysis. In an auto accident, a car hit a pedestrian and the driver then slammed on the brakes to stop the car. During the subsequent trial, the driver's lawyer claimed that the driver was obeying the posted 35 mph speed limit, but that the limit was too high to enable him to see and react to the pedestrian in time. You have been called as the state's expert witness. In your investigation of the accident site, you make the following measurements: The skid marks made while the brakes were applied were 280 ft long, and the tread on the tires produced a coefficient of kinetic friction of 0.30 with the road. (a) In your testimony in court, will you say that the driver was obeying the posted speed limit? You must be able to back up your answer with clear numerical reasoning during cross-examination. (b) If the driver's speeding ticket is \(\$ 10\) for each mile per hour he was driving above the posted speed limit, would he have to pay a ticket, and if so, how much would it be?

Step-by-Step Solution

Answer

(a) No, the driver was speeding at approximately 50.13 mph. (b) The ticket cost is \$151.30.

Step 1: Identify Given Values

We are given that the length of the skid marks is 280 ft, and the coefficient of kinetic friction is 0.30. The speed limit is 35 mph. We need to determine if the car's speed exceeded this limit when the driver began braking.

Step 2: Convert Speed Limit to Feet per Second

First, convert 35 mph to feet per second (ft/s) using the conversion 1 mph = 1.467 ft/s: \[35 \, \text{mph} \times 1.467 = 51.345 \, \text{ft/s}\]

Step 3: Apply Kinematic Equation for Braking

Use the work-energy principle. The work done by friction is equal to the change in kinetic energy:\[- f_k \cdot d = \frac{1}{2} m v_i^2\]where \(f_k = \mu_k \cdot m \cdot g\). Rearrange:\[\mu_k \cdot m \cdot g \cdot d = \frac{1}{2} m v_i^2\]\[2 \cdot \mu_k \cdot g \cdot d = v_i^2\] where \(\mu_k = 0.3\), \(g = 32.2\, \text{ft/s}^2\), and \(d = 280\, \text{ft}\). Solve for initial speed \(v_i\): \[v_i = \sqrt{2 \cdot 0.3 \cdot 32.2 \cdot 280}\] Calculate \(v_i\).

Step 4: Calculate Initial Speed

Plug the values into the equation: \[v_i = \sqrt{2 \cdot 0.3 \cdot 32.2 \cdot 280} = \sqrt{5414.4} \approx 73.56 \, \text{ft/s}\]

Step 5: Convert Initial Speed to Miles per Hour

Convert the initial speed from ft/s to mph:\[73.56 \, \text{ft/s} \div 1.467 \approx 50.13 \, \text{mph}\]

Step 6: Determine Speeding Violation

Compare the found speed with the posted speed limit. If 50.13 mph > 35 mph, then the driver was speeding.

Step 7: Calculate Speeding Ticket Cost

The speeding fine is \$10 per mph over the speed limit. The driver was \(50.13 - 35 = 15.13\) mph over the limit. The ticket cost is:\[15.13 \, \text{mph} \times 10 = 151.3\, \text{dollars}\]

Key Concepts

Work-Energy Principle

The work-energy principle is a powerful tool in physics that relates the work done on an object to its change in kinetic energy. In the context of an auto accident, when a car applies brakes, the work done by kinetic friction (a force that opposes motion) brings the car to a halt. This work is calculated as the product of the frictional force and the distance the car skids.

• The force of kinetic friction, denoted by \( f_k \), is determined using the coefficient of kinetic friction \( \mu_k \) and the normal force, which is usually the weight of the car (mass \( m \) times gravity \( g \)).
• The formula for work done by friction is \(-f_k \cdot d\), where \(d\) is the distance of the skid marks.
• This work equals the initial kinetic energy, \( \frac{1}{2} m v_i^2 \), where \( v_i \) is the initial speed before braking.

Rearranging the work-energy formula provides a method to solve for \( v_i \), allowing us to deduce how fast the vehicle was moving right before it began to decelerate.

Kinematic Equations

Kinematic equations are essential for interpreting the motion of objects under uniform acceleration. In situations like an auto accident, these equations help determine how a vehicle’s speed changes over time and distance. Here, a specific kinematic equation is applied:

• The kinematic equation used is derived from the work-energy principle: \[ v_i = \sqrt{2 \cdot \mu_k \cdot g \cdot d} \]
• This formula allows calculation of the vehicle's speed right before braking (\(v_i\)), with \( \mu_k = 0.3\), \( g = 32.2\, \text{ft/s}^2\), and \( d = 280\, \text{ft} \).

Such an equation simplifies the process of transitioning from dynamic motion (as in a moving car) to stopping (through kinetic friction), ensuring calculations accurately reflect real-world conditions.

Auto Accident Analysis

Analyzing an auto accident requires careful consideration of various factors, such as vehicle speed, road conditions, driver reactions, and braking effectiveness. As an expert witness in a trial context, you utilize physics principles to examine the evidence:

• Skid marks reveal the distance over which the driver attempted to stop.
• The coefficient of kinetic friction indicates how effectively the tires interact with the road.
• By calculating initial speeds and comparing them with speed limits, you assess whether the driver adhered to traffic laws.

Your analysis, based on the detailed measurements and calculations, is vital in legal proceedings to determine the circumstances of the accident and the driver's responsibility.

Speed Calculation

Calculating speed accurately involves converting between different units and understanding their relationships. In this auto accident analysis:

• Speed limits are usually given in miles per hour (mph), while physical laws are often applied in feet per second (ft/s).
• Conversion between these units is crucial: \[ \text{Speed in ft/s} = \text{Speed in mph} \times 1.467 \]
• For example, a speed limit of 35 mph translates to 51.345 ft/s.
• Initial speed calculated during braking (e.g., 73.56 ft/s) is converted back to mph to compare against legal limits.

Speed calculations help determine violations, such as whether a driver exceeded the speed limit and, consequently, if fines are applicable based on how much the speed limit was surpassed.