Problem 105 - Applied Calculus

Question

\(105-106 .\) 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: $$ 150 \text { feet } $$

Step-by-Step Solution

Answer

The estimated speed is approximately 82 mph.

Step 1: Identify the Formula

We need to use the formula given in the problem to estimate the speed of the car. The formula is \( y = 9.4x^{0.37} \), where \( y \) represents the speed in mph and \( x \) is the length of the skidmarks in feet.

Step 2: Substitute the Value of x

Substitute the value of \( x = 150 \) feet into the formula: \( y = 9.4 \times 150^{0.37} \).

Step 3: Calculate the Exponent

Calculate \( 150^{0.37} \). Using a calculator, we find that \( 150^{0.37} \approx 8.7716 \).

Step 4: Multiply by the Coefficient

Multiply the result from the previous step by 9.4: \( 9.4 \times 8.7716 \approx 82.454 \).

Step 5: Round the Result

The calculated speed \( 82.454 \) mph should be rounded as appropriate. Rounding to the nearest whole number gives us \( y \approx 82 \) mph.

Key Concepts

Speed Estimation from Skidmarks

To estimate the speed of a car from its skidmarks, we use the formula: \[ y = 9.4x^{0.37} \] Here, \( y \) is the speed in miles per hour (mph), and \( x \) is the length of the skidmarks in feet. This formula helps us not only gauge the car's speed but also understand how the length of the skidmark relates to the velocity of the vehicle at the start of braking. In the exercise you're tackling, the skidmarks measured 150 feet. By substituting \( x = 150 \) into the formula, the calculation becomes straightforward. But why does this formula work the way it does? The formula incorporates the physics of friction and deceleration before skidding begins, capturing the relationship between speed and the resultant skidmarks. Factors like road conditions may alter the calculations slightly, but this provides a solid approximation.

Skidmarks Analysis

Analyzing skidmarks is crucial in investigations related to road accidents. Skidmarks give investigators observable data to understand how fast a vehicle was traveling at the moment the brakes were applied up until the car came to a complete stop. When examining skidmarks, several factors can influence the length, such as:

• The type of tires and their tread patterns.
• Condition of the road surface, whether dry or wet.
• The braking system's responsiveness and effectiveness.

All these elements play roles in the resulting skid size. The details, collected during the initial inspections, feed into precise formulas to calculate the estimated speed of the vehicle involved. This process ties directly into road safety investigations and assessments by providing a more unified approach to understanding vehicle accidents.

Deceleration Calculation

Deceleration is the rate at which a vehicle slows down. In the context of skidmarks, it represents how quickly the car is stopping from its initial speed. For the basic given formula, the deceleration is already somewhat factored into the equation. However, breaking it down can further help investigators fine-tune their assessments of vehicular speed estimations. When estimating deceleration, consider:

• How effective the braking system worked.
• Environmental factors like road friction which is influenced by conditions such as wetness or the type of road.
• The vehicle's mass, as heavier vehicles typically take longer to decelerate.

Decelerating forces act over the stopping distance to quantify how quickly velocity changes. This is crucial to identifying why skidmarks appeared at specific lengths relative to the initial vehicle speed. By refining the deceleration factors, more accurate speed predictions can be determined when combined with skidmark analysis.