When the side of an open tank filled with water is punctured, water flows out according to Bernoulli's equation: P1+12ρv12+ρgy1=P2+12ρv22+ρgy2where P is the external pressure, ρ is the density of water (10³ kg/m³), v is the flow velocity, g is gravitational acceleration (10 m/s²), and y is the elevation. If water flows out at a rate of 6 m/s, what is the height of the water above the hole?

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11ecba2d_1302_fc6a_a3bf_950aafe5b2f4_MD0008_00 Bernoulli's equation is based on the conservation of energy for fluids. The sum of the energy due to the external pressure (P), movement of the fluid 11ecba2d_1303_237b_a3bf_831f02d05b0b_MD0008_00 , and the weight of the fluid 11ecba2d_1303_237c_a3bf_178de07db82a_MD0008_00 must be equal at any two points. The left side of the equation is set at the surface of the water in the tank (1), and the right side of the equation is set to the hole outside the tank (2): 11ecba2d_1303_237d_a3bf_09ec68ef9237_MD0008_00 Before solving for the height of the water above the hole, the equation can be simplified by canceling out terms. The external pressures P₁ and P₂ are both equal to atmospheric pressure and cancel each other out. The velocity at which the water level falls is much lower than the velocity of the water leaving through the hole (v₁ « v₂), so v₁ and 11ecba2d_1303_237e_a3bf_331d03b61710_MD0008_00 are considered to be negligible and equal to 0. By setting the height of the hole to be the reference elevation, y₂ and 11ecba2d_1303_237f_a3bf_312a9fc4f520_MD0008_00 Therefore, the equation is greatly simplified: 11ecba2d_1303_2380_a3bf_af1ac351af00_MD0008_00 11ecba2d_1303_4a91_a3bf_e7b4d8d09d62_MD0008_00 This equation shows that the potential energy above the hole is equal to the kinetic energy out the hole. Canceling out ρ from both sides and using the given values (g = 10 m/s² and v₂ = 6 m/s), the elevation of the water above the hole is: 11ecba2d_1303_4a92_a3bf_b340b7da7f9b_MD0008_00 11ecba2d_1303_4a93_a3bf_31a826b50b7c_MD0008_00 Educational objective: Bernoulli's equation describes the equilibrium of the pressure, kinetic energy, and potential energy between two points in a flowing liquid. When the equation is applied to a punctured tank, the weight of the water above the hole (potential energy) drives the movement of water through the hole (kinetic energy).

When the Side of an Open Tank Filled with Water