A small crack occurs at the base of a 13.5 m high dam. The effective crack area through which water leaves is 1.55 10-3 m2.
(a) Ignoring viscous losses, what is the speed of water flowing through the crack?
To determine the speed of water flowing through the crack, we can use the principle of conservation of energy. We can equate the potential energy of the water at the top of the dam to its kinetic energy at the bottom, neglecting any energy losses due to viscosity.
The potential energy of the water at the top of the dam is given by the equation:
PE = m * g * h
where PE is the potential energy, m is the mass of the water, g is the acceleration due to gravity, and h is the height of the dam.
The kinetic energy of the water at the bottom is given by the equation:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the water, v is the velocity of the water.
Since the mass of the water cancels out in both equations, we can equate the two equations:
m * g * h = (1/2) * m * v^2
Simplifying, we get:
v = sqrt(2 * g * h)
Substituting the given values, we have:
v = sqrt(2 * 9.8 m/s^2 * 13.5 m)
v ≈ 15.2 m/s
Therefore, the speed of water flowing through the crack is approximately 15.2 m/s.