Highest Water Fountain The world's highest fountain of water is located, appropriately enough, in Fountain Hills, Arizona. The fountain rises to a height of 560 ft
 (5 feet higher than the Washington Monument). How long does it take for water to reach the top of the fountain? 

To find how long it takes for the water to reach the top of the fountain, we need to solve for time in the equation of motion for an object under the influence of gravity. The formula for the height \( h \) reached by an object under constant acceleration (gravity) is given by:

\[
h = v_0 t - \frac{1}{2} g t^2
\]

where:
- \( h = 560 \, \text{ft} \) (the height of the fountain),
- \( v_0 \) is the initial velocity of the water,
- \( g = 32.2 \, \text{ft/s}^2 \) is the acceleration due to gravity,
- \( t \) is the time it takes for the water to reach the top.

At the highest point, the water's velocity will be zero, so we can use a simpler kinematic equation:

\[
v_f^2 = v_0^2 - 2gh
\]

where \( v_f = 0 \) at the top of the fountain. Solving for \( v_0 \) gives:

\[
0 = v_0^2 - 2gh
\]
\[
v_0^2 = 2gh
\]
\[
v_0 = \sqrt{2gh}
\]

Substituting the known values:

\[
v_0 = \sqrt{2 \times 32.2 \, \text{ft/s}^2 \times 560 \, \text{ft}}
\]

Let's calculate \( v_0 \).The initial velocity \( v_0 \) of the water is approximately 189.91 feet per second.

Now, to find the time \( t \) it takes for the water to reach the top, we use the equation:

\[
v_f = v_0 - g t
\]

Since \( v_f = 0 \) at the top, solving for \( t \) gives:

\[
t = \frac{v_0}{g}
\]

Substituting the known values:

\[
t = \frac{189.91 \, \text{ft/s}}{32.2 \, \text{ft/s}^2}
\]

Let’s calculate the time \( t \).The time it takes for the water to reach the top of the fountain is approximately 5.90 seconds.

Would you like more details or have any questions?

Here are 5 related questions to think about:
1. How long does it take for the water to fall back down?
2. What would happen to the time if the fountain's height was doubled?
3. What is the velocity of the water as it reaches halfway up the fountain?
4. How much kinetic energy does the water have at the base of the fountain?
5. How would the time change if the fountain were on another planet with a different gravitational force?

**Tip:** The time for the water to fall back down from the top of the fountain is the same as the time it takes to go up!

The world's highest fountain of water is located, appropriately enough, in Fountain Hills, Arizona. The fountain rises to a height of 560 ft (5 feet higher than the Washington Monument). How long does it take for water to reach the top of the fountain?

Find out how long it takes for water to reach the top of the world's highest fountain, standing 560 feet tall. This problem involves the principles of kinematics, including projectile motion and gravitational acceleration. Suitable for students in Grades 9-12, this detailed step-by-step solution explores velocity, time, and height calculations.

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