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 calculate the speed of water flowing through the crack, you can make use of the principle of conservation of energy.
Here are the steps to find the answer:
Step 1: Identify the potential energy of the water at the top of the dam and the kinetic energy of the water flowing out of the crack.
Potential energy (PE) can be calculated using the formula: PE = mgh, where m is the mass of the water, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the dam (13.5 m in this case).
Kinetic energy (KE) can be calculated using the formula: KE = 0.5mv^2, where m is the mass of the water and v is the speed of the water flowing out of the crack.
Step 2: Set the potential energy equal to the kinetic energy.
Since energy is conserved, we can equate the two expressions from Step 1:
mgh = 0.5mv^2.
Step 3: Simplify the equation.
Mass (m) cancels out on both sides of the equation, giving us:
gh = 0.5v^2.
Step 4: Solve for the speed (v).
Rearrange the equation to isolate v:
v^2 = 2gh.
Finally, take the square root of both sides to find the speed (v):
v = √(2gh).
Step 5: Plug in the values.
Substitute the values into the equation:
v = √(2 × 9.8 m/s^2 × 13.5 m).
Calculating this expression will give you the speed of water flowing through the crack, ignoring viscous losses.