Problem 31 - Calculus with Concepts in Calculus

Question

A racing car accelerated from rest, traveling 2400 feet in 12 seconds. Must the car have been traveling at least 130 miles per hour at some moment during that time interval? Explain. (Hint: Convert to miles and hours.)

Step-by-Step Solution

Answer

Yes, the car traveled at an average speed of approximately 136.5 mph, exceeding 130 mph.

Step 1: Convert Distance to Miles

The car traveled 2400 feet. To convert feet to miles, use the conversion factor of 1 mile = 5280 feet. So, \(\text{distance in miles} = \frac{2400}{5280} \approx 0.4545 \text{ miles}\).

Step 2: Convert Time to Hours

The time taken was 12 seconds. To convert seconds to hours, use the conversion factor of 1 hour = 3600 seconds. So, \(\text{time in hours} = \frac{12}{3600} = 0.00333 \text{ hours}\).

Step 3: Calculate Average Speed

Use the formula for average speed, which is \(\text{average speed} = \frac{\text{distance}}{\text{time}}\). Plug in the distance and time in miles and hours: \(\text{average speed} = \frac{0.4545}{0.00333} \approx 136.5 \text{ miles per hour}\).

Step 4: Compare with Required Speed

The problem asks if the car must have been traveling at least 130 miles per hour at some moment. The average speed of 136.5 miles per hour, which is greater than 130 miles per hour, implies that the car reached a speed of at least 130 miles per hour at some point.

Key Concepts

Unit Conversion

Unit conversion is pivotal when comparing different measurements like speed or distance in various units. In the racing car problem, we started with distances in feet and time in seconds, while we needed to determine speed in miles per hour. Thus, converting units accurately is essential in providing results in a common measure.
To convert the distance of 2400 feet into miles, it is crucial to know that 1 mile equals 5280 feet. Using this conversion factor, the calculation is done by dividing the distance in feet by the number of feet in a mile:

• Distance in miles = \(\frac{2400}{5280} \approx 0.4545\) miles.

Similarly, converting time from seconds to hours requires knowing there are 3600 seconds in one hour. So, for our problem:

• Time in hours = \(\frac{12}{3600} = 0.00333\) hours.

Successfully performing these conversions allows us to derive the average speed in the correct units of miles per hour.

Speed in Physics

In physics, speed is defined as the distance traveled over a unit of time. It is a scalar quantity, meaning it only has magnitude and no directional component. Understanding the formula for average speed is crucial in solving real-world physics problems, like determining how fast a car travels over a certain time period.
The formula for average speed is:

• Average speed = \(\frac{\text{distance}}{\text{time}}\)

In our scenario involving a racing car, we calculated:

• Average speed = \(\frac{0.4545 \text{ miles}}{0.00333 \text{ hours}} \approx 136.5 \text{ mph}\)

The result shows the car's average speed over the entire period, allowing for the comparison with any required speed measures. In this way, speed calculations in physics help us understand various real-world dynamics.

Racing Car Problem

Racing car problems are fascinating applications of physics, requiring calculations to determine speed and acceleration over specific intervals. In these problems, understanding average speed helps confirm whether the car has possibly met certain speed thresholds at any point. In this exercise, we must determine if the car exceeded 130 mph during a 12-second acceleration phase.
By calculating the average speed (136.5 mph) using the earlier unit conversions, we find it is higher than the target speed of 130 mph. This implies that the car must have reached, and likely exceeded, 130 mph at some point, considering the nature of averages in continuous data.
Such problems challenge us to apply unit conversion and speed concepts, confirming how theoretical knowledge plays a pivotal role in practical scenarios like racing.