📘 Concept & Theory Concept Used ›
This question is based on the concepts of distance, displacement, uniform circular motion, and speed.
- Distance is the total length of the actual path travelled by an object.
- Displacement is the shortest straight-line distance between the initial and final positions of an object, along with direction.
- For a circular path, the distance covered in one complete revolution is equal to the circumference of the circle.
- The circumference of a circle is given by: \[ C=\pi d \] where \(d\) is the diameter of the circle.
- After completing one full revolution on a circular track, the athlete returns to the starting point. Therefore, displacement becomes zero.
- After completing half a revolution, the athlete reaches the diametrically opposite point. In this case, displacement is equal to the diameter of the circle.
🗺️ Solution Roadmap Step-by-step Plan ›
Calculate the circumference of the circular track.
Determine the athlete's speed using distance covered in one round and time taken.
Convert the given time into seconds.
Calculate the total distance covered in the given time.
Determine the number of rounds completed.
Find the final position of the athlete and hence calculate displacement.
📊 Graph / Figure Graph / Figure ›
✏️ Solution Complete Solution ›
- Given
- Diameter of circular track: \[ d=200~m \]
- Time taken to complete one round: \[ 40~s \]
- Total time: \[ 2~minutes~20~seconds \]
- Diameter of circular track: \[ d=200~m \]
- Circumference of the circular track: \[ C=\pi d \]
- Substituting the value of diameter: \[ \begin{aligned} C&=\frac{22}{7}\times200\\ &=\frac{4400}{7}\;m \end{aligned} \]
- Therefore, distance covered in one complete round is: \[ \frac{4400}{7}\;m \]
- Time taken to complete one round: \[ t=40~s \]
- Speed of the athlete: \[ \begin{aligned} v&=\frac{\text{Distance}}{\text{Time}}\\ &=\frac{\frac{4400}{7}}{40}\\ &=\frac{4400}{280}\\ &=\frac{110}{7}\;m/s \end{aligned} \]
- Convert the total time into seconds: \[ \begin{aligned} t&=(2\times60)+20\\ &=120+20\\ &=140~s \end{aligned} \]
- Distance covered in 140 seconds: \[ \begin{aligned} \text{Distance}&=v\times t\\ &=\frac{110}{7}\times140\\ &=110\times20\\ &=2200~m \end{aligned} \]
- Hence, the total distance covered by the athlete is: \[ \boxed{2200~m} \]
- Now, calculate the number of rounds completed: \[ \begin{aligned} \text{Number of rounds} &=\frac{\text{Total time}}{\text{Time for one round}}\\ &=\frac{140}{40}\\ &=\frac{7}{2}\\ &=3\frac{1}{2} \end{aligned} \]
- This means the athlete completes:
- 3 complete rounds and returns to the starting point.
- An additional half round, reaching the point diametrically opposite to the starting point.
- The displacement after 3 complete rounds is zero because the athlete is back at the starting position.
- During the additional half round, the athlete reaches the opposite end of the circle.
- Therefore, displacement is equal to the diameter of the circular track: \[ \text{Displacement}=200~m \]
- Hence, \[ \boxed{\text{Displacement}=200~m} \]
💡 Answer Final Answer ›
- Distance covered: \[ \boxed{2200~m} \]
- Displacement: \[ \boxed{200~m} \]
🎯 Exam Significance Exam Significance ›
- Frequently asked to test the difference between distance and displacement.
- Checks understanding of circular motion and shortest-path concepts.
- Students often incorrectly write displacement as zero; therefore it is an important conceptual question.
- Useful for numerical questions involving speed, time, and distance.
🔑 Key Takeaways Key Takeaways ›
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Distance depends on the actual path travelled.
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Displacement depends only on initial and final positions.
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One complete round on a circular track gives zero displacement.
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Half a round on a circular track gives displacement equal to the diameter.
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Distance and displacement can have completely different values.
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In this problem: \[ \text{Distance}=2200~m \] while \[ \text{Displacement}=200~m \]