Water leaks from an open tank via a hole of area 2 mm ^2 at the bottom. Assume the tank is filled with water to a height of 80 cm and has a cross-sectional area of 0.4 m ^2 . The pressure at the open surface and at the hole equals atmospheric pressure. Neglecting the small velocity of water near the open surface of the tank, determine the time required for half of the water to leak out.

Step-by-step solution

Let h be the height of water in the tank at any time t . The velocity of efflux from the hole at the bottom is given by Torricelli's law: v = 2gh The rate of decrease of the volume of water in the tank is equal to the rate of flow of water through the hole. Using the equation of continuity: -A dh dt = a v = a 2gh Rearranging the variables to integrate: dt = - A a 2g h^ -1/2 dh To find the time T required for half of the water to leak out, we integrate from h = H to h = H/2 : ₀^ T dt = - A a 2g _ H ^ H/2 h^ -1/2 dh