The reading of pressure metre attached with a closed pipe is 4.5 × 10⁴ N / m². On opening the valve, water starts flowing and the reading of pressure metre falls to 2.0 × 10⁴ N / m². The velocity of water is found to be √V m / s. The value of V is _________.

Solution

Physics sir · 3 steps

The pressure that disappears when the valve opens has turned into kinetic energy of the water.

1. Same height. The gauge reads the pressure at one fixed point of a horizontal pipe, so the ρ g h term is the same before and after. With the valve closed the water is at rest, so v = 0 there.

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

2. Put in numbers. P₁ - P₂ = 4.5 × 10⁴ - 2.0 × 10⁴ = 2.5 × 10⁴ N m⁻² and ρ = 1000 kg m⁻³ for water. So v² = (2 × 2.5 × 10⁴)/1000 = 50 m² s⁻².
3. So. v = √50 m s⁻¹, which gives V = 50.

The common slip: 25. Dropping the 1/2 in 1/2ρ v² gives v² = 25, not 50.

The idea to keep. A pressure drop at the same height pays for kinetic energy: Δ P = 1/2ρ v². Always use the difference of the two readings, not either reading alone.

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