Question

A horizontal pipe widens from a radius of 0.150 m to 0.280 m. If the speed of the water is 1.60 m/s in the larger pipe, what is the speed in the smaller pipe (in m/s)?

Answer 1

Radius of the larger pipe = R₁ = 0.28 m

Cross-sectional area of the larger pipe = A₁ = R₁²

Speed of the water in the larger pipe = V₁ = 1.6 m/s

Radius of the smaller pipe = R₂ = 0.15 m

Cross-sectional area of the smaller pipe = A₂ = R₂²

Speed of the water in the smaller pipe = V₂

By Continuity Equation,

V₂ = 5.575 m/s

Speed of the pipe in the smaller pipe = 5.575 m/s

Answer 2

Calculation:

1. Identify the given values:

• Radius of smaller pipe, $r_1=0.150\text{m}$

• Radius of larger pipe, $r_2=0.280\text{m}$

• Speed in larger pipe, $v_2=1.60\text{m/s}$

2. Calculate the cross-sectional areas:

• Area of smaller pipe:

  $$A_1=\pi r_1^2=\pi (0.150{)}^2=0.0707{\text{m}}^2$$

• Area of larger pipe:

  $$A_2=\pi r_2^2=\pi (0.280{)}^2=0.246{\text{m}}^2$$

3. Apply the continuity equation ($A_1{v}_1=A_2{v}_2$):

   $$v_1=\frac{A_2{v}_2}{A_1}=\frac{(0.246{\text{m}}^2)(1.60\text{m/s})}{0.0707{\text{m}}^2}$$$$v_1=5.57\text{m/s}$$

The speed in the smaller pipe is 5.57 m/s