Question

A slow jogger runs a mile in \(13 \mathrm{~min} .\) Calculate the speed in (a) in/s, (b) \(\mathrm{m} / \mathrm{min}\), (c) \(\mathrm{km} / \mathrm{h} .(1 \mathrm{mi}=1609 \mathrm{~m}\) \(1 \mathrm{in}=2.54 \mathrm{~cm} .)\)

Short answer

The speed of the jogger is (a) 81.23 in/s, (b) 123.77 m/min and (c) 7.43 km/h.

Step-by-step solution

Convert the Distance Units

We start by changing the distance of 1 mile into the units needed for each part of the question. Using the conversion factors given in the question, 1 mile is equal to 1609 meters, and 1 inch is equal to 2.54 cm, or 0.0254 m. So, the mile can also be converted to inches by dividing 1609 meters by 0.0254 m/in which gives 63360 inches.

Calculate Speed in in/s

The speed is calculated by dividing the distance over time. For part (a), we need the units in inches per second. So we take the 63360 inches and divide it by the time in seconds. Given that there are 60 seconds in a minute, 13 minutes will be \(13 \times 60 = 780\) seconds. Therefore, the speed is \( \frac{63360}{780} = 81.23\) in/s.

Calculate Speed in m/min

For part (b), using the same method, we need to convert the speed in m/min. The distance in meters is 1609 and the time is already in minutes (13 mins). Therefore, the speed in m/min is \( \frac{1609}{13} = 123.77\) m/min.

Calculate Speed in km/h

For part (c), the speed is needed in km/h. We first convert the distance in meters (1609 m) to kilometers by dividing by 1000, giving 1.609 km. We also need to convert the time in minutes to hours by dividing by 60. Therefore, the speed in km/h is \( \frac{1.609}{13/60} = 7.43\) km/h.

Key concepts

Unit Conversion

Understanding unit conversion is essential when calculating measures in different units. It's a process of changing one type of unit to another while maintaining the same quantity value. To perform this, you must multiply or divide by a conversion factor, which relates the two units.

For example, to convert miles to meters, we used the given conversion rate that 1 mile equals 1609 meters. Similarly, for inches to centimeters, we applied that 1 inch is 2.54 centimeters, which is the same as 0.0254 meters when converting to metric units. A thorough knowledge of these conversions is crucial, and having a reference table or conversion calculator handy can be incredibly helpful for students.

Distance and Time

The relationship between distance and time is fundamental in understanding motion. Distance measures how far an object has traveled, and time measures how long it has taken to travel that distance. In the given exercise, the jogger covers a mile in 13 minutes.

To calculate various speed units, one needs to express distance and time in compatible units that match the desired outcome for speed. Sometimes, this means keeping the units as they are, like meters per minute (m/min), and other times it requires converting units, such as converting minutes to hours for kilometers per hour (km/h). It is pivotal to ensure that these units are consistent across the calculation to avoid errors in the final speed value.

Speed Calculation

Speed calculation is a straightforward division of distance by time, but it's pivotal to have both in matching units. In the original exercise, three different speed units were required: inches per second (in/s), meters per minute (m/min), and kilometers per hour (km/h).

Inches per Second (in/s)

After converting 1 mile to 63360 inches and 13 minutes to 780 seconds, speed was found by dividing the two for the result of 81.23 in/s.

Meters per Minute (m/min)

The metric conversion used the direct figures: 1609 meters over 13 minutes, resulting in 123.77 m/min.

Kilometers per Hour (km/h)

The final conversion sought a larger scale, requiring a conversion to 1.609 kilometers and dividing by the time converted to hours (13/60), yielding 7.43 km/h.

This process demonstrates that understanding how to manipulate units in the context of both time and distance is key to accurately determining speed. It's beneficial for students to practice these calculations with multiple unit types to gain confidence in speed calculations across different systems.