Question

The ratio of velocities of water in a pipe lying horizontally at two ends is 1:4. What will be the ratio of diameters of pipe at these two ends?
A) 1:2
B) 2:1
C) 1:4
D) 4:1​

Answer 1

Concept:

Equation of continuity: The continuity equation states that At any given point along the pipe, the cross-sectional area's product with the fluid's velocity is constant.

Given:

The ratio of velocities of water in a pipe at two ends is $1:4$.

Find:

The ratio of diameters of pipe at the two ends.

Solution;

Let the velocity of water and cross-section area of one end be $v_1$ and $A_1$ respectively.

The velocity of water and cross-section area of the other end be $v_2$ and $A_2$ respectively.

According to the equation of continuity,

$A_1v_1=A_2v_2$

$\frac{v_1}{v_2}=\frac{A_2}{A_1}$

The ratios of the velocities is $1:4$.

So,

$\frac{1}{4}=\frac{A_2}{A_1}$

The area of a circle is $\pi r^2$.

$\frac{1}{4}=\frac{\pi (r_2)^2}{\pi (r_1)^2}$

$\frac{1}{4}=\frac{(r_2)^2}{(r_1)^2}$

$\frac{r_2}{r_1}=\frac{1}{2}$

$\frac{r_1}{r_2}=\frac{2}{1}$

The diameter of a circle is twice the radius of the circle.

Therefore, the ratio of the diameters is:

$\frac{2r_1}{2r_2}=\frac{4}{2}$

$\frac{D_1}{D_2} =\frac{2}{1}$

Hence, the ratio of the diameters of pipe at two ends is $2:1$.

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Answer 2

Answer:

The ratio of diameters of the pies at two ends 2:1

Explanation:

Concept:

Equation of continuity: According to the continuity equation, the cross-sectional area multiplied by the fluid's velocity is constant at any given location along the pipe.

Given:

Water flowing through a pipe at its two ends moves at a 1:4 ratio.

Find:

The pipe's ratio of diameters at its two ends.

Let v1 and A1 represent the water's velocity and cross section area of one end, respectively.

The water's flow rate and cross-section area at the other end are, respectively, v2 and A2.

According to the equation of continuity

A₁v₁ = A2v2

v1/v2 = A2/A1

The ratio of the velocity is 1:4

1/4=A2/A1

1/4=(R2)^2/(R1)^2

R2/R1=D2/D1=1/2

The diameter of a circle is twice the radius of the circle.

Therefore, the ratio of the diameters is:

D1/D2=2/1

Hence , the ratio of diameters of the pies at two ends 2:1

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