Problem 98 - Brief Applied Calculus

Question

Police or insurance investigators often want to estimate the speed of a car from the skidmarks it left while stopping. A study found that for standard tires on dry asphalt, the speed (in mph) is given approximately by \(y=9.4 x^{0.37}\), where \(x\) is the length of the skidmarks in feet. (This formula takes into account the deceleration that occurs even before the car begins to skid.) Estimate the speed of a car if it left skidmarks of: 350 feet.

Step-by-Step Solution

Answer

The estimated speed of the car is approximately 117.88 mph.

Step 1: Identify the Given Values

We are given that the skidmark length, \( x \), is 350 feet. We need to estimate the speed \( y \) using the given formula: \( y = 9.4 x^{0.37} \).

Step 2: Substitute the Given Value into the Formula

Substitute \( x = 350 \) into the formula \( y = 9.4 x^{0.37} \). This gives us \( y = 9.4 \times 350^{0.37} \).

Step 3: Calculate the Power of the Skidmark Length

Calculate \( 350^{0.37} \). Using a calculator or mathematical software, we find that \( 350^{0.37} \approx 12.54 \).

Step 4: Calculate the Estimated Speed

Now multiply the result from Step 3 by 9.4: \( y = 9.4 \times 12.54 \). This gives us \( y \approx 117.88 \).

Step 5: Interpret the Result

The estimated speed of the car is approximately 117.88 mph when it left skidmarks of 350 feet on dry asphalt.

Key Concepts

Skidmark Analysis

Skidmark analysis is a technique often used by accident investigators to determine the events leading up to a car accident. Specifically, they study the marks left by the tires on the road to deduce how the vehicle was behaving just before it came to a stop. When a car brakes suddenly, it creates visible marks due to the friction between the tires and the road surface.
These marks provide clues about the car's speed, direction, and braking effectiveness. In forensic analysis, understanding the length and nature of these skidmarks can help reconstruct accident scenes, offering critical insights into the vehicle's speed and the driver's reaction time. For our exercise, these marks play a key role in estimating the car's speed at the time it began to brake.

Speed Estimation

Estimating the speed of a car based on the length of skidmarks involves using mathematical formulas derived from empirical studies. The formula provided, \( y = 9.4 x^{0.37} \),allows us to calculate the approximate speed (\( y \)) of a car by considering the length of the skidmarks (\( x \)).The formula takes into account not only the skid part but also the deceleration that begins even before skidding starts. This is why it's possible to get a reliable estimate of the car's speed at the moment braking began. In our example, the skidmark length was 350 feet, and substituting this into the formula gives us a speed of approximately 117.88 mph.

Deceleration Calculation

Deceleration is the reduction in speed over time, often occurring when a vehicle applies the brakes. It’s a crucial factor in skidmark analysis because it helps determine the forces involved during a vehicle stop. In simple terms, deceleration is the "slowing down" process and is usually measured as a negative acceleration value. The formula for speed estimation includes a component that accounts for deceleration even before the skidmarks appear. This aspect is significant because tires do not start marking the road immediately upon brake application; thus, the car is already slowing down before leaving visible marks. Understanding the deceleration allows investigators to assess the braking efficiency and vehicle behavior prior to the full stop, providing a more comprehensive view of the factors leading up to the incident.