Water is flowing at a speed of `1.5ms^(-1)` through a horizontal tube of cross-sectional area `10^(-2)m^(2)` and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm, the minimum force that you must exert should be (density of water = `10^3 kgm^(-3)).`

Options

A. 15 N
B. 22.5 N
C. 33.7 N
D. 45 N

Answer

To solve the problem of determining the minimum force required to stop the flow of water with a given speed, we can follow these steps:

### Step 1: Identify the initial and final velocities of the water.
- The initial velocity \( v_1 \) of the water is given as \( 1.5 \, \text{m/s} \).
- The final velocity \( v_2 \) after hitting the palm is \( 0 \, \text{m/s} \).

### Step 2: Calculate the change in velocity (\( \Delta v \)).
\[ \Delta v = v_1 - v_2 = 1.5 \, \text{m/s} - 0 \, \text{m/s} = 1.5 \, \text{m/s} \]

### Step 3: Calculate the mass of water flowing per unit time.
To find the mass of water flowing through the tube, we need to use the formula:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]
The volume of water flowing through the tube per second can be calculated using the cross-sectional area and the velocity:
\[ \text{Volume per second} = \text{Area} \times \text{Velocity} = A \times v_1 \]
Given:
- Area \( A = 10^{-2} \, \text{m}^2 \)
- Density \( \rho = 10^3 \, \text{kg/m}^3 \)

Calculating the volume:
\[ \text{Volume} = 10^{-2} \, \text{m}^2 \times 1.5 \, \text{m/s} = 1.5 \times 10^{-2} \, \text{m}^3/s \]

Now, calculating the mass of water flowing per second:
\[ \text{Mass} = \rho \times \text{Volume} = 10^3 \, \text{kg/m}^3 \times 1.5 \times 10^{-2} \, \text{m}^3/s = 15 \, \text{kg/s} \]

### Step 4: Calculate the force using the formula \( F = m \cdot a \).
We can use the formula for force:
\[ F = m \cdot \frac{\Delta v}{\Delta t} \]
Assuming the time interval \( \Delta t = 1 \, \text{s} \) for simplicity:
\[ F = 15 \, \text{kg/s} \cdot \frac{1.5 \, \text{m/s}}{1 \, \text{s}} = 15 \, \text{N} \]

### Final Answer:
The minimum force that you must exert to stop the flow of water is \( \boxed{15 \, \text{N}} \).

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