Hydroelectric power plants convert the gravitational potential energy of falling water into electrical power, typically by allowing the water to flow through a pipe called a penstock to rotate a generator located below it. Let the bottom of the penstock be the origin of a Cartesian coordinate system and the point at which the gravitational potential energy is zero.

h = 61 m

a.) Consider a penstock that is vertical and has a height of h = 61 m. How long, t in seconds, does it take water to fall from the top of the penstock to the bottom? Assume the water starts at rest.

b.)How much total mechanical energy in houles does a mass of 1 kg of water have when it reaches the generator?

Explanation:

In hydroelectric power , potential energy is converted into electrical energy

height given h = 61 m .

acceleration due to gravity g = 9.8

using the relation

h = ut + 1/2 gt²  , t is time taken to reach the bottom .

61 = 1/2 x 9.8 x t²

t² = 12.49

t = 3.53 s

b )

when a mass of 1 kg of water reaches the bottom , its potential energy will be converted into mechanical energy ( kinetic energy )

potential energy = mgh

m = mass of water = 1 kg

acceleration due to gravity g = 9.8

h = 61 m

potential energy = 1 x 9.8 x 61

= 597.8 J .

mechanical energy = 597.8 J .

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min