Problem 97 - 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: 150 feet.

Step-by-Step Solution

Answer

The car's estimated speed is approximately 49.14 mph.

Step 1: Identify given information and formula

Determine what information is given in the problem. We know the length of the skidmarks is 150 feet. The formula to estimate speed is \( y = 9.4x^{0.37} \), where \( x \) is the length of the skidmarks.

Step 2: Plug the length of the skidmarks into the formula

Substitute \( x = 150 \) into the given formula: \( y = 9.4 \times 150^{0.37} \). This substitution will allow us to compute the speed of the car.

Step 3: Calculate the exponent

Compute \( 150^{0.37} \). This is the part of the formula where we raise the skidmark length to the power of 0.37. Using a calculator, \( 150^{0.37} \approx 5.228 \).

Step 4: Multiply by the constant

Multiply the result of the previous step by 9.4: \( y = 9.4 \times 5.228 \). This multiplication will give us the estimated speed of the car.

Step 5: Complete the calculation

Finally, perform the multiplication: \( y = 9.4 \times 5.228 \approx 49.1432 \). Hence, the estimated speed of the car is approximately 49.14 mph.

Key Concepts

Speed Estimation

Estimating the speed of a car using skidmarks is a fascinating aspect of applied calculus. It involves using mathematical relationships to derive information about real-world events. In our example, we have a formula that relates skidmark length to the speed of a car. When we say speed estimation, we mean calculating how fast a car was traveling before it started to skid. This is valuable information for traffic accident investigations.

The formula used, \(y = 9.4x^{0.37}\), is a mathematical model that allows us to estimate the speed based on the length of the skidmarks. This specific equation takes into account factors like deceleration and the frictional resistance of tires on asphalt. By utilizing this formula, we can interpret the physical evidence left at the scene to estimate how fast a vehicle was going prior to braking.

Skidmarks Analysis

Skidmarks analysis is crucial for understanding vehicular accidents. Skidmarks are the tire marks left on a road when a driver brakes suddenly and the tires slide over the surface of the road. The length of these marks can provide insight into the car's behavior before and during the accident.

In our scenario, the length of the skidmarks measured was 150 feet. Investigators use the formula \(y = 9.4x^{0.37}\) to turn this measurement into an estimated speed. Here, the variable \(x\) represents skidmark length, and it is raised to the power of 0.37 to model the non-linear relationship between skid length and speed. This exponent is important because it reflects how longer skidmarks do not directly translate to proportionally higher speeds, due to the effects of things like road texture and tire condition.

Mathematical Modeling

Mathematical modeling plays a pivotal role in solving real-world problems like estimating vehicle speeds from skidmarks. Modeling involves creating equations that simulate real-world situations, like the travel of a car leading to an accident.

The equation \(y = 9.4x^{0.37}\) exemplifies a successful mathematical model. This model considers both the physics of motion and empirical data collected from similar vehicular incidents. Models like this are continuously tested and refined through experimental data to ensure they provide high accuracy in diverse conditions.

In practice, using mathematical models allows detectives and accident reconstructionists to gather insights that might not be immediately obvious from physical observations alone. These insights can then be used in reports, legal proceedings, or for educational purposes where learning about physics and calculus principles is the aim.