Correct Bernoulli's equation is (symbols have their usual meaning) :

We need to identify the correct form of Bernoulli's equation.

Bernoulli's equation applies to steady, incompressible, non-viscous fluid flow along a streamline. It is derived from the work-energy theorem (or equivalently, conservation of energy per unit volume).

The three terms represent pressure energy, gravitational potential energy, and kinetic energy — all per unit volume:

$$P + \rho g h + \frac{1}{2}\rho v^2 = \text{constant}$$

where $$P$$ is the fluid pressure, $$\rho$$ is the density, $$g$$ is gravitational acceleration, $$h$$ is the height, and $$v$$ is the fluid velocity.

Option (1) uses mass $$m$$ instead of density $$\rho$$ — incorrect (that would give energy, not energy per unit volume).

Option (2): $$P + \rho gh + \frac{1}{2}\rho v^2 = \text{constant}$$ — this is the correct form.

Option (3) is missing the $$\frac{1}{2}$$ in the kinetic term — incorrect.

Option (4) has an extra $$\frac{1}{2}$$ in the gravitational term — incorrect.

The correct answer is Option (2).