A fountain reaches a height of 140 m. The pumps provide 0.50 m^3/s of water. What is the radius of the jet at the bottom of the fountain? Hint: find the velocity first, then use continuity.
To find the radius of the jet at the bottom of the fountain, we can use the principle of continuity, which states that the mass flow rate of a fluid remains constant as it flows through different sections of a pipe or tube.
First, let's calculate the velocity of the water as it leaves the pump. We can use the equation for the mass flow rate (Q) of a fluid:
Q = A * V
where Q is the mass flow rate, A is the cross-sectional area of the pipe or jet, and V is the velocity of the fluid.
Given that the pumps provide 0.50 m^3/s of water, we can assume that this is the volume flow rate, which can be considered equivalent to the mass flow rate (since the density of water remains constant). Therefore, Q = 0.50 m^3/s.
Now, let's find the cross-sectional area of the jet. Since the jet forms a circle at the bottom of the fountain, the area can be calculated using the formula A = π * r^2, where r is the radius of the jet.
We can rearrange the equation for the mass flow rate to solve for the velocity:
V = Q / A
Substituting the given values:
V = 0.50 m^3/s / (π * r^2)
Now, we can use the principle of continuity. According to the principle, the mass flow rate of the fluid is constant at all points along its path. Since the water leaves the pump and travels upwards in the fountain, the velocity decreases, while the cross-sectional area increases (due to the expanding shape of the fountain). Therefore, we can equate the mass flow rate at these two points:
Q = A * V
0.50 m^3/s = A_bottom * V_bottom
Since the cross-sectional area A_bottom is given by A_bottom = π * r_bottom^2, we can substitute it into the equation:
0.50 m^3/s = π * r_bottom^2 * V_bottom
Now, we have two equations:
V = 0.50 m^3/s / (π * r^2)
0.50 m^3/s = π * r_bottom^2 * V_bottom
By equating the two expressions for the mass flow rate, we can solve for the radius of the jet at the bottom of the fountain (r_bottom).