To solve this question, you can apply Bernoulli's equation to relate the energy (pressure, velocity, and height) of the water at different points. Since you're given the height difference between the reservoir and the tank, and the pressure difference between the two, Bernoulli’s principle is appropriate to find the velocity of water entering the tank.

Step-by-Step Approach:

Bernoulli's equation states:

$\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2$

Where:

• $P_1$, $P_2$ = pressures at points 1 (reservoir) and 2 (tank),
• $v_1$, $v_2$ = velocities at points 1 and 2,
• $z_1$, $z_2$ = elevations at points 1 and 2,
• $\rho$ = density of water,
• $g$ = acceleration due to gravity.

Assumptions for your case:

1. Reservoir (Point 1): The water is at rest, so $v_1 = 0$.
2. Tank (Point 2): The velocity of the water entering the tank is $v_2$, which we need to find.
3. The pressure at the tank is $P_2 = -70 \, \text{kN/m}^2$, which is below atmospheric pressure.
4. The height difference is $z_1 - z_2 = 6 \, \text{m}$.

The simplified equation:

Using Bernoulli's equation between the reservoir and the tank:

$\frac{P_1}{\rho g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2$

Since the water is flowing freely, you can neglect $P_1$ because it's a reference point, and assume the velocity at the reservoir is zero. The equation reduces to:

$\frac{P_2}{\rho g} + z_2 = \frac{v_2^2}{2g} + z_1$

Solving for $v_2$:

$\frac{v_2^2}{2g} = z_1 - z_2 + \frac{P_2}{\rho g}$

$v_2 = \sqrt{2g \left( z_1 - z_2 + \frac{P_2}{\rho g} \right)}$

This formula will allow you to compute the velocity of water entering the tank.

Parameters:

• $g \approx 9.81 \, \text{m/s}^2$ (acceleration due to gravity),
• $P_2 = -70 \, \text{kN/m}^2 = -70,000 \, \text{N/m}^2$ (pressure below atmospheric),
• $\rho = 1000 \, \text{kg/m}^3$ (density of water),
• $z_1 - z_2 = 6 \, \text{m}$ (height difference).

Would you like me to calculate the velocity using this formula, or would you like more details on how to proceed?