In a certain gravity free space, the piston of an injection is being pushed so that the water jet comes out with a speed v. The area of the piston is much greater than the orifice of the injection.

Options

A. The force required to be applied on the piston is proportional to `v^(2)`.
B. The power developed by the force pushing the piston is proportional to `v^3`.
C. The time for emptying the injection is proportional to `v^(-1)`.
D. The total work done in emptying the injection is proportional to `v^2`.

Explanation

To solve the problem, we will analyze the situation step by step, applying the principles of fluid dynamics and mechanics.

### Step 1: Understand the System
We have a piston in a cylinder that is pushing water through an orifice. The area of the piston (A) is much greater than the area of the orifice (a). The water exits the orifice with a speed \( v \).

### Step 2: Apply the Continuity Equation
According to the continuity equation, the flow rate must be constant throughout the system. Therefore, we can write:

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