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 need to use the principle of conservation of energy.

The potential energy of the water at the top of the dam is converted into kinetic energy as it flows through the crack. Assuming no other losses, the potential energy at the top of the dam is equal to the kinetic energy at the bottom of the dam.

The potential energy of an object is given by the equation:

PE = mgh

Where:
PE: Potential Energy
m: Mass
g: Acceleration due to gravity (approximately 9.8 m/s²)
h: Height

In this case, we can consider the mass of the water as its volume multiplied by its density. The density of water is roughly 1000 kg/m³.

The volume of water flowing through the crack per second can be calculated by multiplying the crack area by the speed of water flowing through it.

Let's calculate the speed of water flowing through the crack step by step:

1. Calculate the potential energy at the top of the dam:
PE_top = m * g * h
PE_top = (1.55 * 10^-3 m²) * (1000 kg/m³) * (9.8 m/s²) * (13.5 m)

2. Calculate the kinetic energy at the bottom of the dam (which is equal to the potential energy at the top):
KE_bottom = PE_top
KE_bottom = (1.55 * 10^-3 m²) * (1000 kg/m³) * (9.8 m/s²) * (13.5 m)

3. Now, equate the kinetic energy at the bottom of the dam to 1/2 * m * v², where v is the speed of water flowing through the crack.

(1/2) * m * v² = KE_bottom

Solving for v:
v = √((2 * KE_bottom) / m)

Plug in the calculated values to find the speed of water flowing through the crack.