Problem 60 - Algebra 1

Question

An accident reconstructionist is responsible for finding how fast cars were going before an accident. To do this, a reconstructionist uses the model below where \(S\) is the speed of the car in miles per hour, \(d\) is the length of the tires' skid marks in feet, and \(f\) is the coefficient of friction for the road. Car speed model: \(S=\sqrt{30 d f}\) a. In an accident, a car makes skid marks 74 feet long. The coefficient of friction is \(0.5 .\) A witness says that the driver was traveling faster than the speed limit of 45 miles per hour. Can the witness's statement be correct? Explain your reasoning. b. How long would the skid marks have to be in order to know that the car was traveling faster than 45 miles per hour?

Step-by-Step Solution

Answer

In part (a), the speed of the car is calculated and compared to the speed limit to validate or discard the witness's claim. In part (b), the length of the skid marks necessary for the car to be found traveling faster than 45 mph is determined. Final conclusions are drawn based on these calculations.

Step 1: Calculate the speed of the car in the first scenario

The witness statement needs to be confirmed using the speed model \(S=\sqrt{30 d f}\). Given that \(d = 74\) feet and \(f = 0.5\), substituting these values into the model gives \(S = \sqrt{30*74*0.5}\). This should be compared to the given speed limit of 45 mph.

Step 2: Compare calculated speed to speed limit

Once the speed has been calculated, it can be compared to the speed limit. If the calculated speed is greater than the speed limit, then the witness's statement can be taken as correct.

Step 3: Calculate the length of the skid marks for a speed exceeding the speed limit

The speed \(S\) is given as greater than 45 mph. Hence the inequality \(45 < \sqrt{30 d f}\) can be squared and hence rearranged to solve for \(d\). The value of \(f = 0.5\) is substituted into this inequality to find \(d\).

Step 4: Determine if a higher speed is possible

The calculated \(d\) can be compared to any physical or practical limitations for the size of the skid marks. If achieving such a \(d\) is deemed unlikely or impossible, then a conclusion can be drawn that the car could not have been traveling faster than 45 mph.

Key Concepts

Quadratic Equations

Quadratic equations are essential in various mathematical models where relationships between quantities are non-linear. Typically, they have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In real-world scenarios, they can model things like projectile paths or, as in the accident reconstruction exercise, the relationship between skid mark length, friction, and speed.
\(S = \sqrt{30df}\) is not exactly a quadratic equation, but solving for different values leads to squaring both sides, creating an equation with the squared term, similar to a quadratic equation.
To find one of the variables when given others, manipulating the equation often involves algebraic techniques used for quadratics, like isolating terms or leveraging square roots.

Mathematical Modeling

Mathematical modeling translates real-world situations into mathematical expressions, providing a systematic method for understanding and solving problems. It involves identifying relevant variables and relationships, then expressing these as equations or formulas.
In our scenario, the model \(S = \sqrt{30 d f}\) represents the relationship between skid marks, friction, and speed. It simplifies a complex accident scene into measurable and computable terms, allowing for precise speed calculations.
Effective modeling requires knowing the assumptions behind formulas, such as constant coefficient of friction and applying them correctly based on given data. These models are powerful tools in fields like physics, economics, and engineering, providing critical insights and guiding decisions.

Problem Solving

Problem solving in algebra requires a structured approach of understanding the problem, devising a plan, and executing that plan with specific strategies. In our exercise, this involves:

• Identifying and assessing known variables, like skid mark length and friction coefficient.
• Using the speed model \(S = \sqrt{30df}\) to calculate the car's speed.
• Comparing results with given conditions, such as the 45 mph speed limit.

With this methodical approach, one can logically deduce whether a witness's account of an accident is consistent with the evidence. Solving such problems hones critical thinking and analytical skills, which are valuable across various fields.

Speed Calculation

Calculating speed from physical evidence like skid marks is crucial in accident reconstruction. The speed model \(S = \sqrt{30df}\) allows for determining the speed based on measurable evidence from the accident scene.
Steps involved in speed calculation include:

• Substitute the skid mark length \(d\) and friction coefficient \(f\) into the formula.
• Perform the mathematical operations to compute \(S\), which gives the speed in mph.
• Validate the result by comparing it to known limits or thresholds, like speed limits.

This process not only aids in verifying witness statements but also in understanding the dynamics of vehicle movement during an accident, leading to more accurate reconstructions.