Calculating Athlete Speed on a Circular Track

The problem asks us to find the speed of an athlete in kilometers per hour (km/h) who runs around a circular field. We are given the radius of the field, the number of rounds the athlete completes, and the total time taken.

Understanding the Given Information

• Radius of the circular field, \(r = 7\) m
• Number of times the athlete runs around the field = 8
• Total time taken = 3 minutes 40 seconds
• Value of \(\pi = \frac{22}{7}\)

Step 1: Calculate the Distance of One Round (Circumference)

The distance covered in one round is equal to the circumference of the circular field. The formula for the circumference of a circle is \(C = 2 \pi r\).

Using the given values:

Circumference \(C = 2 \times \frac{22}{7} \times 7\) m

\(C = 2 \times 22\) m

\(C = 44\) m

So, the distance covered in one round is 44 meters.

Step 2: Calculate the Total Distance Covered by the Athlete

The athlete runs 8 times around the field. The total distance covered is the distance of one round multiplied by the number of rounds.

Total Distance = Number of rounds \(\times\) Circumference

Total Distance = \(8 \times 44\) m

Total Distance = 352 m

The athlete covers a total distance of 352 meters.

Step 3: Convert Total Time to Seconds

The time is given in minutes and seconds. To calculate speed, it's easier to have the time in a single unit, like seconds.

Total time = 3 minutes 40 seconds

1 minute = 60 seconds

3 minutes = \(3 \times 60 = 180\) seconds

Total time in seconds = 180 seconds + 40 seconds = 220 seconds

The total time taken is 220 seconds.

Step 4: Calculate Speed in Meters per Second (m/s)

Speed is calculated as Total Distance divided by Total Time.

Speed \( = \frac{\text{Total Distance}}{\text{Total Time}} \)

Speed \( = \frac{352 \text{ m}}{220 \text{ s}} \)

We can simplify this fraction:

Speed \( = \frac{352}{220} = \frac{176 \times 2}{110 \times 2} = \frac{176}{110} = \frac{88 \times 2}{55 \times 2} = \frac{88}{55} \)

Both 88 and 55 are divisible by 11:

Speed \( = \frac{88 \div 11}{55 \div 11} = \frac{8}{5} \) m/s

The speed of the athlete is \(\frac{8}{5}\) m/s.

Step 5: Convert Speed from m/s to Kilometers per Hour (km/h)

The question asks for the speed in km/h. To convert a speed from meters per second (m/s) to kilometers per hour (km/h), we multiply by the conversion factor \(\frac{18}{5}\).

Speed in km/h \( = \text{Speed in m/s} \times \frac{18}{5} \)

Speed in km/h \( = \frac{8}{5} \times \frac{18}{5} \)

Speed in km/h \( = \frac{8 \times 18}{5 \times 5} \)

Speed in km/h \( = \frac{144}{25} \) km/h

The speed of the athlete is \(\frac{144}{25}\) km/h.

Summary of Calculations

Conclusion on Athlete Speed

Based on the calculations, the speed of the athlete is \(\frac{144}{25}\) km/h.

Revision Table: Key Concepts for Speed Calculation

Additional Information: Speed, Distance, and Time Basics

Speed, distance, and time are fundamental concepts in physics and mathematics used to describe motion.

• Speed: It is the rate at which an object changes its position. It tells us how much distance is covered in a certain amount of time. The standard unit for speed is meters per second (m/s), but kilometers per hour (km/h) is also commonly used, especially for vehicles and longer distances.
• Distance: It is the total length of the path traveled by an object. In this problem, the path is circular, and the total distance is the sum of the lengths of all the rounds completed.
• Time: It is the duration over which the motion occurs. It is crucial to use consistent units for time when calculating speed. Commonly used units are seconds, minutes, and hours.

Understanding how to convert between different units of speed (like m/s and km/h) is essential for solving many problems related to motion.