Problem 39 - Algebra 1: Concepts and Skills

Question

In Exercises 38 and \(39,\) use the following information. When a car skids to a stop, its speed \(S\) (in miles per hour) before the skid can be modeled by the equation \(S=\sqrt{30 d f} .\) where \(d\) is the length of the tires' skid marks (in feet) and \(f\) is the coefficient of friction for the road. In an accident, a car makes skid marks that are 147 feet long. The coefficient of friction is 0.4. A witness says that the driver was traveling under the speed limit of 35 miles per hour. Can the witness’s statement be correct? Explain your reasoning.

Step-by-Step Solution

Answer

The witness's statement is incorrect. According to the skid marks and the coefficient of friction, the car was going over the speed limit.

Step 1: Understand the Problem

The problem provides a formula for calculating the speed of a skidding car based on the length of skid marks and the road's coefficient of friction. The problem also provides the length of skid marks and coefficient of friction. The goal is to determine whether the witness’s statement about the car not exceeding the speed limit is correct or not.

Step 2: Substitute provided values into the equation

The speed formula is \(S=\sqrt{30 d f},\) where \(d = 147\) feet is the length of the skid marks, \(f = 0.4\) is the coefficient of friction. Substituting these values into the equation yields \(S = \sqrt{30 * 147 * 0.4}.\)

Step 3: Solve the Equation

To find the speed we square root the product of 30, 147 and 0.4, so the speed \(S = \sqrt{30 * 147 * 0.4} = 36.06\) miles per hour.

Step 4: Compare the speed limit

Now we need to compare the calculated speed, 36.06 mph, with the speed limit of 35 mph. As 36.06 mph > 35 mph, we can conclude that according to the skid marks and the coefficient of friction, the car was traveling over the speed limit.

Key Concepts

Skid Marks

Skid marks are lines left on the road when a vehicle's tires lock and slide due to sudden braking. These marks occur because the tires stop rotating, while the car continues to move due to inertia. The length of these skid marks can tell us a lot about the speed the car was traveling before it came to a halt. When analyzing skid marks, a longer mark usually indicates a higher speed before braking. The distance alone provides crucial data about how fast a car came to a stop, which, combined with other factors like road conditions and slope, can help calculate the original speed. In accident reconstruction, experts often use these details to reconstruct the events leading to a collision.

• Longer skid marks indicate potentially higher speeds.
• They are a vital part of accident analysis.
• Helpful in determining pre-accident conditions.

Thus, skid marks not only serve as evidence in accidents but also aid in understanding and calculating speed.

Coefficient of Friction

The coefficient of friction is a value that describes the slip resistance between two surfaces in contact. In this case, it refers to the interaction between a car's tires and the road surface. This factor greatly affects how a car can stop or skid. A higher coefficient of friction means better grip, allowing shorter skid marks for a car traveling at the same speed, indicating better stopping power due to increased traction. Conversely, a lower coefficient indicates less grip, leading to longer skid marks. In our scenario, the coefficient of friction is given as 0.4. This means the road surface does not provide significant grip. When considering a dry asphalt road normally has a coefficient between 0.7 to 0.8, 0.4 suggests perhaps the road was either wet or otherwise slippery.

• A low coefficient means less traction and longer skid marks.
• A high coefficient ensures better stopping ability.
• This value is crucial in calculating stopping distances and speeds.

Speed Calculation

To calculate the speed of a car based on skid marks, we use a specific equation that accounts for both the length of the skid marks and the coefficient of friction. The formula for this is given by:\[ S = \sqrt{30 d f} \]Where:

• \(S\) is the speed in miles per hour,
• \(d\) is the skid mark length in feet, and
• \(f\) is the coefficient of friction.

In our case:

• The skid mark length \(d = 147\) feet.
• The coefficient of friction \(f = 0.4\).

Plugging these parameters into the formula:\[ S = \sqrt{30 \times 147 \times 0.4} = \sqrt{1764} = 36.06 \text{ mph} \]This result indicates the speed at which the vehicle was traveling before it skidded to a stop. Since the calculated speed is 36.06 mph, it is compared to the speed limit of 35 mph. Therefore, it shows the driver was indeed over the speed limit, contrary to the witness's statement.

• This formula helps reconstruct the speed at the moment of braking.
• Allows analysis of whether a car exceeded a speed limit before stopping.
• Is essential in verifying eyewitness accounts in accident scenarios.