A stone is dropped from a height of 5 km. The distance it falls through varies…
2025
A stone is dropped from a height of 5 km. The distance it falls through varies directly with the square of the time taken to fall through that distance. If it falls 64 m in 4 seconds, find the distance the stone covers in the 5th second?
Answer: A. 36 m — When a quantity D varies directly with the square of another quantity T, the relationship has the form D = kT2 for some constant k. The distance covered…
- A.
36 m
- B.
58 m
- C.
100 m
- D.
72 m
Attempted by 10 students.
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Correct answer: A
When a quantity D varies directly with the square of another quantity T, the relationship has the form D = kT2 for some constant k.
The distance covered specifically during the nth second of such a motion is the difference between the cumulative distance up to time n and the cumulative distance up to time (n − 1) — it is not the cumulative distance itself.
Applying this to the stone:
Since the distance varies directly with the square of the time, write D = kT2.
Use the given data (D = 64 m at T = 4 s) to find k: 64 = k × 42 = 16k, so k = 4.
Find the cumulative distance at T = 5 s: D(5) = 4 × 52 = 4 × 25 = 100 m.
The cumulative distance at T = 4 s is the given value, 64 m.
Distance covered during the 5th second = D(5) − D(4) = 100 − 64 = 36 m.
Cross-check: for D = kT2, the distance covered in the nth second always equals k(2n − 1), since D(n) − D(n−1) = k[n2 − (n−1)2] = k(2n − 1). With k = 4 and n = 5, this gives 4 × (2×5 − 1) = 4 × 9 = 36 m — confirming the result.