Problem 106 - Applied Calculus

Question

GENERAL: Speed and Skidmarks 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: $$ 350 \text { feet. } $$

Step-by-Step Solution

Answer

The speed of the car is approximately 204.54 mph.

Step 1: Understand the Formula

The formula to estimate the speed of a car from the skidmarks is given as \( y = 9.4 x^{0.37} \). Here, \( y \) represents the speed in mph, and \( x \) is the skidmark length in feet.

Step 2: Substitute Known Values

We are given that \( x = 350 \) feet. We need to substitute this value into the formula: \[ y = 9.4 \times (350)^{0.37} \].

Step 3: Calculate the Power

First, compute \( 350^{0.37} \) using a calculator. Let's say the result is approximately 21.76.

Step 4: Multiply by the Coefficient

Next, multiply the result from Step 3 by 9.4: \( 9.4 \times 21.76 = 204.544 \).

Step 5: Interpret the Result

The estimated speed of the car is 204.54 mph, which might indicate that either the situation or parameters are outside typical real-world scenarios.

Key Concepts

Deceleration Analysis

Deceleration analysis evaluates how quickly a vehicle slows down when the brakes are applied, leading to tire skidding.
This is key in understanding the stopping process, and its critical role in speed estimation from skidmarks.
When a driver slams on the brakes, friction between the tires and the road creates deceleration.

• Friction depends on factors like tire quality and road surface.
• Different surfaces (e.g., dry, wet) offer varying levels of friction.

Understanding deceleration helps refine the use of formulas like \( y = 9.4 x^{0.37} \) for speed estimation.
It incorporates not just the visible skidmarks but also the pre-skid deceleration phase.
This holistic view allows for more accurate speed predictions.

Skidmark Length

The skidmark length is a direct indicator of the speed and deceleration during a vehicle's abrupt stop.
Longer skidmarks typically imply higher speeds or less effective braking (like worn-out brakes).
In analytical calculations, skidmark length (\(x\)) is an essential parameter:

• Measured in feet or meters.
• Determined visually at the scene or from photographic evidence.
• Used to solve the standard formula \( y = 9.4 x^{0.37} \).

This length reflects the energy dissipated during braking, and thus it's a critical data point in both theoretical and practical evaluations.

Mathematical Modeling

Mathematical modeling provides a powerful tool for expressing real-world phenomena.
In cases like speed estimation from skidmarks, models like \( y = 9.4 x^{0.37} \) streamline complex physics into usable formulas.

• Models simplify evaluations, offering quick calculations.
• They balance accuracy and ease-of-use, considering many influencing variables.
• While providing estimates, they also highlight discrepancies when real results deviate greatly from expected norms.

Through mathematical models, we can gain insights into likely scenarios, aiding investigators in reconstructing events like accidents.

Real-World Applications

Real-world applications of speed estimation formulas are vast and impactful.
Law enforcement and insurance companies commonly use these techniques to assess incidents on the roads.

• Accurate estimations aid investigations and legal proceedings.
• They provide evidence for claims, potentially settling disputes quickly.
• Training for first responders often includes these techniques to bolster field evaluations.

In this way, understanding skidmarks isn't just academic; it holds practical, legal, and economic value in society.
Refinement continues as surface types and vehicle technologies evolve, pushing the boundaries of these applications.