Water is flowing through a tube of radius r with a speed v. If this tube is joined to another tube of radius r/2, what is the speed of water in the second tube?

Options

A. ` v//4 `
B. `v //2 `
C. `2v `
D. `4 v `

Explanation

To solve the problem of water flowing through two tubes of different radii, we can use the principle of conservation of mass, often referred to as the equation of continuity. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the equation of continuity
The equation of continuity states that the product of the cross-sectional area (A) and the velocity (v) of a fluid remains constant along a streamline. Mathematically, this can be expressed as:
\[ A_1 v_1 = A_2 v_2 \]
where:
- \( A_1 \) is the area of the first tube,
- \( v_1 \) is the velocity of water in the first tube,
- \( A_2 \) is the area of the second tube,
- \( v_2 \) is the velocity of water in the second tube.

### Step 2: Calculate the cross-sectional area of both tubes
1. For the first tube with radius \( r \):
\[ A_1 = \pi r^2 \]

2. For the second tube with radius \( \frac{r}{2} \):
\[ A_2 = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} \]

### Step 3: Set up the equation using the equation of continuity
Using the areas and velocities, we can write:
\[ A_1 v_1 = A_2 v_2 \]
Substituting the areas:
\[ (\pi r^2) v = \left(\pi \frac{r^2}{4}\right) v' \]

### Step 4: Simplify the equation
We can cancel \( \pi r^2 \) from both sides:
\[ v = \frac{r^2}{4} v' \]

### Step 5: Solve for \( v' \)
Rearranging the equation to solve for \( v' \):
\[ v' = 4v \]

### Final Answer
The speed of water in the second tube is:
\[ v' = 4v \]

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