Water flows in a horizontal pipe whose one end is closed with a valve. The reading of the pressure gauge attached to the pipe is P₁. The reading of the pressure gauge falls to P₂ when the valve is opened. The speed of water flowing in the pipe is proportional to

Solution

Physics sir · 3 steps

Closed valve means water at rest; open it, and the pressure it loses becomes kinetic energy.

1. Valve closed. With the valve shut, the water in the pipe is at rest. The gauge reads P₁ and the speed is zero.
2. Valve open. Once the valve opens, water flows with speed v and the gauge drops to P₂. The pipe is horizontal, so the ρ g h term is the same in both cases and cancels.

   P₁ + 0 = P₂ + 1/2ρ v² ⇒ v = √(2(P₁ - P₂)/ρ)

3. So. Since ρ is constant, v ∝ √(P₁ - P₂), which is option (b).

The idea to keep. In Bernoulli problems the pressure drop pays for kinetic energy, and kinetic energy goes as v². So speed always goes as the square root of the pressure difference.

Answer from the NTA final answer key. Solution by PrepWiser.