Water is filled in a uniform container of area of cross section A. A hole of cross section area a (≪ A) is made in the container at a height of 20 m above the base. Water streams out and hits a small block placed at some distance from container. With what speed (in m s⁻¹) should the block be moved such that water stream always hits the block? (Given a/A = 1/20.) (Take g = 10 m s⁻².)
Physics · Mixed Chapter
JEE Mains

Question

Water is filled in a uniform container of area of cross section A. A hole of cross section area a (≪ A) is made in the container at a height of 20 m above the base. Water streams out and hits a small block placed at some distance from container. With what speed (in m s⁻¹) should the block be moved such that water stream always hits the block? (Given a/A = 1/20.) (Take g = 10 m s⁻².)

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let h be the instantaneous height of water above the hole. The horizontal efflux velocity is u=√(2gh), while the flight time to the ground is √(2H/g), with H=20 m. Step 2: The horizontal range is x=u√(2H/g)=2√(Hh). Step 3: Volume conservation gives dh/dt=−(a/A)√(2gh). Step 4: Differentiate range: dx/dt=√(H/h) dh/dt=−(a/A)√(2gH)=−(1/20)√400=−1 m s⁻¹. The block must move towards the container at speed 1 m s⁻¹.