Problem 65 - College Physics

Question

The Channel Tunnel, or "Chunnel," which runs under the English Channel between Great Britain and France, is 31 mi long. (There are actually three separate tunnels.) A shuttle train that carries passengers through the tunnel travels with an average speed of \(75 \mathrm{mi} / \mathrm{h}\). On average, how long, in minutes, does the shuttle take to make a one-way trip through the Chunnel?

Step-by-Step Solution

Answer

The shuttle takes approximately 25 minutes.

Step 1: Collect Given Information

We are given the following information: - Length of the Channel Tunnel: 31 mi - Average speed of the shuttle train: 75 mi/h We need to find out how long it takes for the shuttle to travel 31 miles at an average speed of 75 mi/h, expressed in minutes.

Step 2: Use the Speed Formula

The formula connecting distance, speed, and time is:\[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]Using the values given:\[\text{Time} = \frac{31 \text{ mi}}{75 \text{ mi/h}}\]

Step 3: Calculate the Time in Hours

Perform the division to find the time in hours:\[\text{Time} = \frac{31}{75} \approx 0.4133 \text{ hours}\]

Step 4: Convert Time to Minutes

Since there are 60 minutes in an hour, convert the time from hours to minutes:\[0.4133 \text{ hours} \times 60 \text{ minutes/hour} = 24.8 \text{ minutes}\]

Step 5: Round to the Nearest Whole Minute

Since time is usually expressed in whole numbers for practical use, round 24.8 minutes to the nearest whole number. Thus, the time it takes is approximately 25 minutes.

Key Concepts

Speed Calculation

In physics, speed is a fundamental concept that helps us understand how fast an object is moving over a certain period of time. It is a scalar quantity, meaning it only has magnitude and not direction. To calculate speed, you need both the distance an object has traveled and the time it took to travel that distance. The formula to compute speed is given by:

• Speed = Distance / Time

The speed of an object tells us how much distance it covers in a unit of time, such as miles per hour (mi/h) or kilometers per hour (km/h). For example, in the exercise above, the shuttle train travels at an average speed of 75 mi/h. This means it covers 75 miles in one hour under normal conditions.
Understanding speed calculations is crucial in various real-world applications, from computing travel times to planning routes and evaluating transportation efficiency.

Distance-Time Relationship

The relationship between distance and time is central to solving problems involving speed. The distance-time relationship is expressed using the equation:

• Time = Distance / Speed

This formula shows us that time is directly proportional to distance when speed is constant. If you want to travel a longer distance at a constant speed, you will need more time. Conversely, with a shorter distance, you require less time. In our example, the shuttle train travels a distance of 31 miles through the Channel Tunnel.
To calculate the time it takes, we divide the distance by the speed (31 mi by 75 mi/h), resulting in 0.4133 hours. Time is typically more intuitive when expressed in minutes, which leads us to the next concept.

Unit Conversion

Unit conversion is often necessary to express the results in a more practical or understandable way. In physics, we frequently convert between different units, such as hours to minutes or miles to kilometers, to suit the context.
To convert time from hours to minutes, you use the conversion factor of 60 minutes per hour. So, when you have a time of 0.4133 hours, multiplying by 60 gives:

• 0.4133 hours × 60 minutes/hour ≈ 24.8 minutes

Rounding 24.8 minutes to the nearest whole number yields approximately 25 minutes.
This conversion is essential for understanding the time in terms that are more relevant to everyday experiences, like scheduling or planning events. Mastering unit conversions can simplify complex calculations and improve communication by presenting information in familiar units.