Question

Water from a tap emerges vertically downwards with an initial speed of 1.0 m/s. The cross-sectional area of tap is 10^–4 m². Assume that the pressure is constant throughout and that the flow is steady, the cross-sectional area of stream 0.15 m below the tap is:

• A. (a) 5.0 × 10^–4 m²
• A. (b) 1.0 × 10^–4 m²
• A. (c) 5.0 × 10^–5 m²
• A. (d) 2.0 × 10^–5 m²

Answer 1

(c)

From conservation of energy $v_{2}^{2}$ =$v_{1}^{2}$+ 2gh … (1)

[can also be found by applying Bernouilli’s theorem between 1 and 2]

From continuity equation A₁ v₁ = A₂ v₂

v₂ =$\left( \frac{A_{1}}{A_{2}} \right)$v₁ … (2)

Substituting value of v₂ from Eq. (2) in Eq. (1)

$\frac{A_{1}^{2}}{A_{2}^{2}}$.$v_{1}^{2}$ =$v_{1}^{2}$+ 2gh

or $A_{2}^{2}$ =$\frac{A_{1}^{2}v_{1}^{2}}{v_{1}^{2} + 2gh}$

\ A₂ =$\frac{A_{1}v_{1}}{\sqrt{v_{1}^{2} + 2gh}}$

Substituting the given values

A₂ =$\frac{(10^{- 4}m^{2})(1.0m/s)}{\sqrt{(1.0m/s)^{2} + 2(10)(0.15)}}$

A₂ = 5.0 × 10^–5 m²