A vessel completely filled with water has two holes 'P' and 'Q' at depths ' 2 h ' and ' 8 h ' from the top respectively. Hole ' P ' is square of side ' a ' and hole ' Q ' is a circle of radius ' r '. The water flowing out per second from both the holes is same, then side ' a ' of hole ' P ' is

Step-by-step solution

Torricelli's Law states that the velocity of efflux from an orifice at depth h is v = 2gh . The volume flow rate is given by A v . For hole P at depth 2h , the velocity is v_P = 2g(2h) = 2 gh . Its area is A_P = a^2 . For hole Q at depth 8h , the velocity is v_Q = 2g(8h) = 4 gh . Its area is A_Q = r^2 . Equating the volume flow rates: a^2 2 gh = r^2 4 gh Dividing both sides by 2 gh gives: a^2 = 2 r^2 Thus, a = r 2 , corresponding to option B .