In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato. The other potatoes are arranged 3 m apart in a straight line, with a total of 10 potatoes. A competitor starts from the bucket, picks up the nearest potato, runs back to the bucket to drop it in, then returns to pick up the next potato. This process continues until all the potatoes are in the bucket. Based on the above information, answer the following questions :
Question: 1
What is the distance covered to pick up the first potato and drop it in bucket ?
Show Hint
Remember to double the distance for every potato because of the "to-and-fro" movement.
Step 1: Understanding the Concept:
The competitor has to travel to the potato and return to the bucket. The distance covered is twice the displacement of the potato from the bucket. Step 3: Detailed Explanation:
The first potato is 5 m from the bucket.
Distance covered to pick it up \( = 5 \text{ m} \).
Distance covered to drop it back \( = 5 \text{ m} \).
Total distance \( = 5 + 5 = 10 \text{ m} \). Step 4: Final Answer:
The distance covered for the first potato is 10 m.
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Question: 2
What is the distance covered to pick up the second potato and drop it in bucket ?
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The sequence of distances for individual potatoes forms an Arithmetic Progression (AP) with common difference \( d = 2 \times 3 = 6 \text{ m} \).
Step 1: Understanding the Concept:
Each subsequent potato is 3 m further than the previous one. Step 3: Detailed Explanation:
Distance of 1st potato from bucket \( = 5 \text{ m} \).
Distance of 2nd potato from bucket \( = 5 + 3 = 8 \text{ m} \).
Total distance for 2nd potato \( = 2 \times 8 \text{ m} = 16 \text{ m} \). Step 4: Final Answer:
The distance covered for the second potato is 16 m.
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Question: 3
What is the total distance the competitor has to run ?
Show Hint
Alternatively, calculate the sum of distances to the potatoes \( \sum = (5 + 8 + 11 + \dots) \) and then double the final result.
Step 1: Understanding the Concept:
The distances covered for each potato form an Arithmetic Progression. We need to find the sum of the first 10 terms. Step 2: Key Formula or Approach:
Sum of AP: \( S_n = \frac{n}{2} [2a + (n-1)d] \) Step 3: Detailed Explanation:
First term (\(a\)) \( = 10 \text{ m} \).
Second term \( = 16 \text{ m} \).
Common difference (\(d\)) \( = 16 - 10 = 6 \text{ m} \).
Number of potatoes (\(n\)) \( = 10 \).
\[ S_{10} = \frac{10}{2} [2(10) + (10-1)6] \]
\[ S_{10} = 5 [20 + 9 \times 6] \]
\[ S_{10} = 5 [20 + 54] \]
\[ S_{10} = 5 \times 74 \]
\[ S_{10} = 370 \text{ m} \] Step 4: Final Answer:
The total distance the competitor has to run is 370 m.
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Question: 4
If average speed of competitor is 5 m/s, then find the average time taken by competitor to put all the potatoes in the bucket.
Show Hint
Ensure units are consistent (meters and meters/second) before dividing.
Step 1: Understanding the Concept:
Time taken is the ratio of total distance covered to the average speed. Step 2: Key Formula or Approach:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] Step 3: Detailed Explanation:
Total distance covered \( = 370 \text{ m} \) (from part iii-a).
Average speed \( = 5 \text{ m/s} \).
\[ \text{Time} = \frac{370}{5} \]
\[ \text{Time} = 74 \text{ s} \] Step 4: Final Answer:
The average time taken is 74 seconds.
