A tank is filled with water to a height H. A small hole is punched in one of the walls at a depth h below the water's surface. The water leaves the hole in a horizontal direction. The stream of water strikes the floor at a distance x, as shown. Neglecting viscosity, you can calculate the value of x from A) x = H - h B) x = [2h(H - h)]^1/2 C) x = h D) x = 2[h(H - h)]^1/2 E) None of these is correct.

Question

Quizplus · Accepted Answer

The horizontal velocity of the water leaving the hole can be found from Bernoulli's equation:

P + 1/2ρv^2 + ρgh = constant

Assuming the pressure outside the tank and the velocity of the water at the surface of the tank are both zero, the equation simplifies to:

1/2ρv^2 = ρgh

where v is the horizontal velocity of the water leaving the hole, ρ is the density of water, and g is the acceleration due to gravity.

Solving for v, we get:

v = √(2gh)

The time it takes for the water to travel horizontally from the hole to the floor can be found from the horizontal distance x and the horizontal velocity v:

t = x/v

The vertical distance the water falls during that time can be found from:

h' = 1/2gt^2 = h

where h' is the distance the water falls vertically from the hole to the floor.

Therefore, the horizontal distance x is related to the heights H and h by:

x = 2h(H - h) / √(2gh)

Answer D is the closest choice to this expression.

A Tank Is Filled with Water to a Height H