Water flows over a section of Niagara Falls at the rate of \( 1,200 \times 10^{6} \mathrm{~kg} / \mathrm{s} \) and falls 50.00 m . How much power is generated by the falling water?

To calculate the power generated by the falling water, we can use the formula for gravitational potential energy and the relationship between power, energy, and time. The potential energy (PE) can be calculated as: \[ PE = mgh \] where: - \( m = 1,200 \times 10^{6} \mathrm{~kg} \) (mass flow rate), - \( g = 9.81 \mathrm{~m/s^2} \) (acceleration due to gravity), - \( h = 50.00 \mathrm{~m} \) (height). Using the values given, we calculate: \[ PE = (1,200 \times 10^{6} \text{ kg}) \times (9.81 \text{ m/s²}) \times (50.00 \text{ m}) = 5.886 \times 10^{9} \text{ J/s} \] Since power is the rate at which energy is generated or used over time, and in this case, we consider energy generated every second, the power \( P \) is equal to the potential energy calculated: \[ P = 5.886 \times 10^{9} \text{ W} \] Thus, the power generated by the falling water is approximately \( 5.89 \times 10^{9} \text{ watts} \) or \( 5.89 \text{ GW} \). Power generated by the falling water at Niagara Falls is truly a staggering number. Just imagine the energy that can be harnessed from such natural forces! This kind of power output contributes significantly to the electricity generated in areas like Niagara, showcasing the immense potential of hydroelectric energy. The sheer volume of water cascading down symbolizes a renewable source that's clean, efficient, and immensely powerful!