Problem 81 - A Graphical Approach to College Algebra

Question

Solve each problem. To estimate the speed \(s\) at which a car was traveling at the time of an accident, a police officer drives a car like the one involved in the accident under conditions similar to those during which the accident took place and then skids to a stop. If the car is driven at 30 miles per hour, the speed \(s\) at the time of the accident is given by $$s=30 \sqrt{\frac{a}{p}}$$ where \(a\) is the length of the skid marks and \(p\) is the length of the marks in the police test. Find \(s\) if \(a=900\) feet and \(p=97\) feet.

Step-by-Step Solution

Answer

The speed \(s\) was approximately 91.38 miles per hour.

Step 1: Substitute Given Values

Start by substituting the given values of \(a = 900\) and \(p = 97\) into the formula \(s = 30 \sqrt{\frac{a}{p}}\). This gives us: \(s = 30 \sqrt{\frac{900}{97}}\).

Step 2: Calculate the Fraction

Calculate the fraction \(\frac{900}{97}\). This gives a value of approximately \(9.2784\).

Step 3: Compute the Square Root

Take the square root of the value obtained in Step 2. \(\sqrt{9.2784} \approx 3.046\).

Step 4: Multiply by 30

Multiply the result from Step 3 by 30 as given in the formula: \(s = 30 \times 3.046\). This results in approximately \(s = 91.38\).

Key Concepts

Skid Marks

Skid marks provide crucial evidence in the analysis of vehicular accidents. These marks are the dark streaks left on the road when a vehicle's tires slide uncontrollably over the surface. By examining these skid marks, particularly their length, investigators can gain insight into the behavior of the vehicle before coming to a halt. When a car comes to a sudden stop, it transfers momentum into frictional heat, resulting in these visible marks. The length of skid marks helps determine the speed at which a vehicle was traveling during an incident. Longer marks typically indicate a higher speed prior to braking. Understanding skid marks is fundamental because it allows law enforcement to estimate traveling speed accurately, helping to reconstruct the events leading up to the accident. In our exercise, the skid marks from the accident scene and those recorded during a police test are compared. This helps to identify the deceleration characteristics of the involved vehicle, hence allowing an accurate speed estimation.

Police Test

After an accident, one of the methodologies used to estimate the speed of a car is conducting a police test. In this test, officers recreate the conditions of the accident as closely as possible with a similar vehicle. During this simulation, they intentionally skid the car to a stop to produce reference skid marks. By comparing these test skid marks to those found at the accident site, police can more accurately determine the speed of the vehicle involved. The length of the skid marks from the police test, denoted in our problem as 'p', acts as a baseline for calculations. By maintaining similar road conditions, tire grip, and vehicle weight, the police test allows for closer approximations in estimating the speed that the accident vehicle must have been traveling. This kind of empirical testing ties the real-world physical evidence collected at the scene with mathematical calculations to provide evidence in traffic accident reconstructions.

Algebraic Formula

The algebraic formula given in the exercise is crucial for estimating the speed of the vehicle. It reads as: \[ s = 30 \sqrt{\frac{a}{p}} \]where:

• \(s\) is the speed of the vehicle at the time of the accident.
• \(a\) is the length of the skid marks at the accident scene.
• \(p\) is the length of the skid marks from the police test.

The formula incorporates a constant multiplier (30 in this case) which accounts for units and factors like road conditions and vehicle specifications. The central element of the formula is the square root of the ratio \(\frac{a}{p}\). This ratio compares the length of skid marks from the actual accident to those from a controlled test, providing a multiplicative factor to adjust the known test speed to estimate the speed for the actual event. The algebraic manipulation of substituting given values into this formula directly enables the calculation of the vehicle's speed at the time of the accident.

Step-by-Step Solution

Breaking down the given formula's application into a step-by-step solution helps understand the process:_1. **Substitute Given Values** First, input the given values into the formula: \( a = 900 \) and \( p = 97 \). This gives us the expression \(s = 30 \sqrt{\frac{900}{97}}\).2. **Calculate the Fraction** Divide 900 by 97 to find the numerical value of the fraction, resulting in approximately 9.2784.3. **Compute the Square Root** Find the square root of 9.2784, which is roughly 3.046. This step reflects how the comparative length of skid marks is translated into a speed factor.4. **Multiply by 30** Finally, multiply the square root by 30 (in our formula, the given constant), yielding the estimated speed: 91.38 mph. By following these steps meticulously, you apply mathematical principles of substitution and computation to solve the problem, ensuring an accurate and reliable speed estimation.