Water level is maintained in a cylindrical vessel placed on horizontal floor up to a fixed height H. A tiny hole of area A is punched in the side wall at a height y from the bottom of the vessel. The emerging stream strikes the ground at a horizontal distance y (≠0) from cylinder. To maintain the level in the vessel at H, the rate (volume/sec) of addition of water is A√(2gH/n). Find n.
Physics · Mixed Chapter
JEE Mains

Question

Water level is maintained in a cylindrical vessel placed on horizontal floor up to a fixed height H. A tiny hole of area A is punched in the side wall at a height y from the bottom of the vessel. The emerging stream strikes the ground at a horizontal distance y (≠0) from cylinder. To maintain the level in the vessel at H, the rate (volume/sec) of addition of water is A√(2gH/n). Find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: At a hole height y from the bottom, the water depth above the hole is H−y. Therefore the efflux speed is v=√[2g(H−y)]. Step 2: Maintaining water level requires inflow rate Q=A v=A√[2g(H−y)]. Step 3: Comparing Q=A√(2gH/n) gives n=H/(H−y). Step 4: The printed statement that the jet lands a horizontal distance y away supplies 2√[y(H−y)]=y (for y≠0), which implies y=4(H−y), i.e. y=4H/5. Then n=H/(H/5)=5.