A water tank springs a leak. Find the speed of water emerging from the hole if the leak is 2.5 m below the surface of the water, which is open to the atmosphere.
4m/s
To find the speed of water emerging from the hole, you can use the concept of Bernoulli's equation, which states that the sum of the pressure energy, kinetic energy, and potential energy is constant along a streamline.
In this case, the water is open to the atmosphere, so the pressure at the hole is atmospheric pressure. We can assume that the flow of water is steady and incompressible, so the kinetic energy remains constant.
Let's break down the steps to find the speed of water emerging from the hole:
Step 1: Determine the pressure at the hole.
Since the water is open to the atmosphere, the pressure at the hole is atmospheric pressure. We can represent it as P_0.
Step 2: Calculate the potential energy at the hole.
The potential energy at the hole is determined by the height of the water, which is 2.5 m. We can represent it as h.
Step 3: Apply Bernoulli's equation.
Bernoulli's equation states that:
P + 1/2 * ρ * v^2 + ρ * g * h = constant
where P is the pressure at the hole, ρ is the density of water, v is the velocity of water emerging from the hole, and g is the acceleration due to gravity.
Considering that P = P_0 and rearranging the equation, we get:
1/2 * ρ * v^2 = P_0 - ρ * g * h
Step 4: Solve for v.
We can rearrange the equation from step 3 to solve for v:
v = √((2 * (P_0 - ρ * g * h)) / ρ)
Step 5: Substitute known values and calculate.
Substitute the known values into the equation from step 4 and calculate the speed of water emerging from the hole.
- Density of water (ρ) = 1000 kg/m^3
- Acceleration due to gravity (g) = 9.8 m/s^2
- Atmospheric pressure (P_0) = 101325 Pa (standard atmospheric pressure)
- Height (h) = 2.5 m
Calculating the velocity will give you the speed of water emerging from the hole.
To find the speed of water emerging from the hole in the water tank, we can use Bernoulli's equation, which relates the pressure, height, and velocity of a fluid.
Bernoulli's equation is given as:
P + 0.5 * ρ * v^2 + ρ * g * h = constant
where P is the pressure, ρ is the density of the fluid, v is the velocity of the fluid, g is the acceleration due to gravity, and h is the height above a reference point (in this case, the surface of the water).
In this scenario, the water is open to the atmosphere, so we can consider the pressure at the surface of the water to be atmospheric pressure, denoted as P_atm. The pressure at the hole in the tank can be calculated using the hydrostatic pressure formula:
P_hole = P_atm + ρ * g * h
Since the water is emerging from the hole, the velocity of the water (v) at the hole is 0. Therefore, the Bernoulli equation simplifies to:
P_atm + ρ * g * h = P_hole
Solving for the velocity, v:
v = √(2 * g * h)
Given that the leak is 2.5 m below the surface of the water, we can substitute the values into the equation to find the velocity:
v = √(2 * 9.8 m/s^2 * 2.5 m)
v ≈ 7.85 m/s
Therefore, the speed of water emerging from the hole in the water tank is approximately 7.85 m/s.