A tank, placed on the ground, is filled with water up to a height ℎ. A small hole is made at a height ℎ1 such that ℎ1 < ℎ. The water jet emerging from the hole strikes
the ground at a horizontal distance D, as shown schematically in the figure. Which of the following statements is correct? (g is the acceleration due to gravity)
the ground at a horizontal distance D, as shown schematically in the figure. Which of the following statements is correct? (g is the acceleration due to gravity)
- Velocity at ℎ1 is \(\sqrt{2𝑔ℎ_1}\)
- 𝐷 = 2(ℎ − ℎ1)
- D will be maximum when ℎ1 = \(\frac{2}{3}\) ℎ
- The maximum value of D is h
The Correct Option is D
Solution and Explanation
The water jet’s horizontal distance \( D \) from the hole to the point where it strikes the ground depends on the velocity of the jet at height \( h_1 \) and the gravitational pull. The velocity at the hole is given by:
\[ v = \sqrt{2gh_1} \]
This is the velocity of the jet when it leaves the hole at height \( h_1 \), as the velocity is due to the gravitational potential energy being converted into kinetic energy.
Key Observation:
The maximum horizontal distance \( D \) occurs when the jet’s flight time and horizontal speed are maximized.
Conclusion:
The maximum horizontal distance \( D \) is greatest when the height of the hole is maximized, i.e., \( h_1 = h \). Therefore, the maximum value of \( D \) is:
\[ D = h \]
Thus, the maximum value of \( D \) is indeed \( h \).
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