Problem 32 - Calculus with Concepts in Calculus

Question

In the aftermath of a car accident it is concluded that one driver slowed to a halt in 9 seconds while skidding 400 feet. If the speed limit was 30 miles per hour, can it be proved that the driver had been speeding? (Hint: 30 miles per hour is equal to 44 feet per second.)

Step-by-Step Solution

Answer

Yes, the driver was speeding.

Step 1: Understand the Given Information

We are given that a driver skidded to a halt over 400 feet in 9 seconds and need to determine if they were speeding. Importantly, the speed limit is given as 30 miles per hour, which is equivalent to 44 feet per second.

Step 2: Calculate Initial Speed

We will use the formula for average speed during deceleration: \( v_{avg} = \frac{d}{t} \), where \( d \) is the distance (400 feet) and \( t \) is the time (9 seconds). Calculating gives \( v_{avg} = \frac{400}{9} \approx 44.44 \) feet per second. In uniform deceleration, initial speed \( v_i = 2 \times v_{avg} = 2 \times 44.44 = 88.88 \) feet per second.

Step 3: Compare Initial Speed to Speed Limit

Now, compare the initial speed with the speed limit. The speed limit is 44 feet per second, while the initial speed was calculated as 88.88 feet per second.

Step 4: Conclusion

Since the initial speed of the driver (88.88 feet per second) is significantly higher than the speed limit (44 feet per second), it can be concluded that the driver was speeding at the time of the accident.

Key Concepts

Deceleration

Deceleration is the process by which a moving object reduces its velocity. In the context of the car accident problem, the car slows down until it comes to a complete stop. This process is crucial for determining if the driver was speeding. During deceleration, the average speed of the car can be significantly less than the initial speed. It's important to measure how quickly this deceleration happens as it affects road safety and accident analysis.
In the provided problem, the car decelerates uniformly over a distance of 400 feet in 9 seconds. "Uniformly" means the rate of deceleration is constant, which simplifies calculations. This slowing down can be represented mathematically, helping us understand the vehicle's behavior leading up to the accident. Using formulas like average speed can help determine the original speed before deceleration began. If the vehicle stopped over a shorter distance with less time, it implies a higher speed before deceleration started.

Average Speed Calculation

Average speed calculation is a key step in analyzing the car accident situation. It helps us find out how fast the car was going on average while it was skidding. We use the formula for average speed: \[ v_{avg} = \frac{d}{t} \] where \( d \) is the distance and \( t \) is the time.
With the car skidding 400 feet over 9 seconds, the average speed becomes approximately 44.44 feet per second. This doesn't yet tell us the driver's speed at the moment before they began to slow down, because this average speed represents the entire skidding event. • The initial speed \( v_i \) can be found using another formula: \( v_i = 2 \times v_{avg} \) Calculating this gives us \( v_i \approx 88.88 \) feet per second. We derive this from the knowledge that with uniform deceleration, the initial speed is roughly twice the average speed. So, the precise calculation hinges on understanding this crucial point.

Speed Limit Comparison

To determine if the driver was speeding, comparing the calculated speed to the speed limit is essential.
Here's how we do it:

• The speed limit given in the problem is 30 miles per hour, which converts to 44 feet per second. This conversion is important for direct comparison since our speed calculations are in feet per second.
• With the initial speed calculated as 88.88 feet per second, we see a significant difference. The driver's speed is over twice the allowed limit of 44 feet per second.

This comparison leaves little room for doubt—the initial speed being much higher than the speed limit confirms that the driver was indeed speeding. Such comparisons are vital in accident investigations, where understanding speed behaviors can explain how accidents occur and help in assessing responsibilities.