The World's Tallest Fountain
When I was in Jeddah in Saudi Arabia, I was told that the city had the world's tallest fountain. I imagined that it might be a couple hundred feet tall, but when I asked how tall it was, and I was completely shocked by the answer: it was nearly three times as tall as the Statue of Liberty!
King Fahd's Fountain is a fountain in Jeddah that shoots seawater to a height of approximately 260 meters, or 850 feet! We went to visit the fountain one evening, and I was just blown away by its height. It's nearly as tall as the Eiffel Tower. Another thing I found crazy was that I had expected a fountain like this to be relatively new, considering the technology it must have taken to create. However, the fountain was actually opened in 1985, over thirty years ago!
So when I saw the fountain, my first question was how fast must the water be shot out of the fountain to reach such a height? It would be a difficult problem to solve if we take air resistance and changing wind speeds into account, but if we ignore these things, it's not actually that difficult. Also, let's assume the water is shot out at ground level to make calculations easier.
We know that a projectile has a height function given by f(x) = -4.9x^2 + vx, measured in meters per second, where v is the initial velocity of the water. We know that the maximum height of a parabola in the form y = ax^2 + bx + c occurs at x = -b/(2a) = -v/(-9.8)=v/9.8. So the maximum height will be given by h = f(v/9.8) = -4.9(v/9.8)^2 + v^2/9.8 ≈ 0.051 v^2. So the relationship is approximately given by h = 0.051 v^2, or v = sqrt(19.6h).
Because we know that the height achieved must be 260 meters, we evaluate this when h = 260 to get v = 71 meters per second.
Sample Problems
1. Calculate the water speed necessary for the fountains at the Bellagio Hotel in Las Vegas, which reaches a maximum height of 460 feet. Ignore air resistance.
2. Calculate the water speed necessary for the fountains at the Burj Khalifa in Dubai, which reaches a maximum height of 500 feet. Ignore air resistance.
3. Because of air resistance, we know that the actual water speeds will be different. In fact, Wikipedia says that the water speed of King Fahd's Fountain is 375 km/h, which converts to about 104 m/s. Compare this to our calculation. Is it higher or lower than our calculation? Why should this be the case?