Problem 115 - Fundamentals of Physics

Question

In \(1889,\) at Jubbulpore, India, a tug-of-war was finally won after \(2 \mathrm{~h} 41 \mathrm{~min},\) with the winning team displacing the center of the rope \(3.7 \mathrm{~m} .\) In centimeters per minute, what was the magnitude of the average velocity of that center point during the contest?

Step-by-Step Solution

Answer

The average velocity is approximately 2.30 cm/min.

Step 1: Convert Time to Minutes

The total time for the tug-of-war contest is 2 hours and 41 minutes. First, convert the time into minutes: \(2 \text{ hours} = 2 \times 60 = 120 \text{ minutes}\), then add the 41 minutes. So the total time in minutes is \(120 + 41 = 161 \text{ minutes}\).

Step 2: Convert Displacement to Centimeters

The displacement of the rope is given in meters. Convert the 3.7 meters displacement into centimeters since the velocity unit requested is in centimeters per minute. We know 1 meter equals 100 centimeters, so \(3.7 \text{ meters} = 3.7 \times 100 = 370 \text{ centimeters}\).

Step 3: Calculate the Average Velocity

Average velocity is calculated by dividing the total displacement by the total time. Using the converted values, the average velocity is: \[\text{Average Velocity} = \frac{\text{Total Displacement in centimeters}}{\text{Total Time in minutes}} = \frac{370 \text{ cm}}{161 \text{ min}}\approx 2.30 \text{ cm/min}\].

Step 4: Round Off the Result

To ensure clarity and precision, we often round our final results to two decimal places when dealing with such contexts unless asked otherwise. Therefore, the magnitude of the average velocity, rounded to two decimal places, is approximately \(2.30 \text{ cm/min}\).

Key Concepts

Unit Conversion

Understanding unit conversion is essential in various fields of science and math, making problems easier to solve by keeping units consistent. In the tug-of-war problem, both displacement and time need to be converted to the requested units before calculating average velocity.

• Length conversion: In this example, we convert meters to centimeters because the desired unit for velocity is in centimeters per minute. The basic conversion factor to remember is that 1 meter equals 100 centimeters. Therefore, to convert 3.7 meters to centimeters, you multiply 3.7 by 100, resulting in 370 centimeters.


• Time conversion: While often, seconds to minutes is a more common conversion, here, you convert hours to minutes. Noting that 1 hour is equal to 60 minutes enables us to convert any hours to minutes quickly.

Always ensure that you multiply by the correct conversion factor to avoid inaccuracies in your calculations, which can significantly affect the final result.

Displacement

Displacement measures how far and in which direction an object has moved from its initial position. It is a vector quantity, meaning it has both magnitude and direction. However, in the context of this problem, only the magnitude (or size) of the displacement is used.

Here, the displacement of the rope in the tug-of-war is 3.7 meters, which is straightforward since the contest's linear nature requires basic length measurement. For calculations, it is essential to convert this measurement into centimeters because the problem asks for velocity in centimeters per minute. Converting this gives a new displacement value of 370 centimeters.

In practical scenarios, problems may involve both horizontal and vertical components of displacement, but here it is a simple linear movement, minimizing complexity.

Time Conversion

Time conversion is another crucial aspect when solving physics problems to keep calculations precise. In this exercise, you are asked to find the average velocity over time expressed in minutes, not hours and minutes.

• Conversion from hours and minutes: The original time duration given is 2 hours and 41 minutes. To convert this into a pure minute measurement, you start by converting hours into minutes. Multiplying 2 hours by 60 gives 120 minutes. Then, you simply add the remaining 41 minutes."
• Adding the values: Adding these two results, 120 minutes + 41 minutes, gives a total of 161 minutes.

Understanding time conversions will provide a consistent time unit, an essential step when calculating velocity or speed in physics problems, allowing accurate comparison or analysis.