A plate moves normally with the speed `v_(1)` towads a horizontal jet of uniform area of cross-section. The jet discharge water at the rate of volume `V` per second at a speed of `v_(2)`. The density of water is `rho`. Assume that water splashes along the surface of the plate ar right angles to the original motion. The magnitude of the force action on the plate due to the jet of water is

Options

A. `rhoVv_(1)`
B. `rho(V)/(v_(2))(v_(1)+v_(2))^(2)`
C. `(pV)/(v_(1)_v_(2))(v_(1))^(2)`
D. `pV(v_(1)+v_(2))`

Explanation

To solve the problem, we need to determine the magnitude of the force acting on the plate due to the water jet. We can approach this step by step.

### Step 1: Understand the scenario
The plate is moving towards a horizontal jet of water with speed \( v_1 \). The jet discharges water at a speed \( v_2 \) and at a rate of volume \( V \) per second. The water splashes off the plate at right angles to its original motion.

### Step 2: Identify the relevant physics concepts
The force exerted by the water jet on the plate can be calculated using the principle of momentum. The force is equal to the rate of change of momentum of the water hitting the plate.

### Step 3: Write the formula for force
The force \( F \) can be expressed as:
\[ F = \frac{dP}{dt} \]
where \( P \) is the momentum.

### Step 4: Define momentum
Momentum \( P \) is defined as:
\[ P = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity.

### Step 5: Determine the change in momentum
The change in momentum \( \Delta P \) for the water hitting the plate can be calculated. Since the water is moving towards the plate with speed \( v_2 \) and the plate is moving towards the water with speed \( v_1 \), the effective change in velocity when the water splashes off the plate is:
\[ \Delta v = v_1 + v_2 \]

### Step 6: Calculate the mass flow rate
The mass flow rate of water can be expressed in terms of density \( \rho \) and the volume flow rate \( V \):
\[ \text{Mass flow rate} = \rho \cdot V \]

### Step 7: Substitute into the force equation
Substituting the mass flow rate and the change in velocity into the force equation gives:
\[ F = \text{mass flow rate} \cdot \Delta v = \rho V (v_1 + v_2) \]

### Step 8: Final expression for force
Thus, the magnitude of the force acting on the plate due to the jet of water is:
\[ F = \rho V (v_1 + v_2) \]

### Conclusion
The force acting on the plate due to the jet of water is given by the formula:
\[ F = \rho V (v_1 + v_2) \]