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?

Short 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-by-step solution

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.

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.

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 \(d\).

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.