The world's highest fountain of water is located appropriately enough in the Fountain Hills, Arizona. The fountain rises to a height of 560 ft ( 5 feet higher than the Washington Monument). a. what is the initial speed of water? b how long does it take for water to reach the top of the fountain?
189.3 m/s
5.9s
To calculate the initial speed of the water and the time it takes to reach the top of the fountain, we can use the equations of motion. Let's assume that the water is launched vertically upward with an initial speed.
a. To find the initial speed of the water, we can use the equation:
v = u + at,
where v is the final velocity (which is zero at the top of the fountain), u is the initial velocity (unknown), a is the acceleration (due to gravity), and t is the time taken to reach the top.
At the top of the fountain, the final velocity is zero, so the equation becomes:
0 = u - gt,
where g is the acceleration due to gravity, which is approximately 9.8 m/s².
Now, we need to convert the height of the fountain to meters, as the acceleration due to gravity is given in meters per second squared:
Height of the fountain = 560 ft = 560 ft × 0.3048 m/ft ≈ 170.688 m.
Using this information, we can solve for the initial speed:
0 = u - (9.8 m/s²)t.
Since the initial speed is what we're looking for, we can rearrange the equation:
u = (9.8 m/s²)t.
b. To calculate the time it takes for the water to reach the top of the fountain, we can use the equation for vertical displacement:
s = ut + (1/2)at².
At the top of the fountain, the displacement is equal to the height of the fountain:
s = 170.688 m.
Substituting the values into the equation, we get:
170.688 = u(t) + (1/2)(-9.8)(t)².
Simplifying further:
170.688 = (9.8 m/s²)t + (1/2)(-9.8 m/s²)(t)².
This equation can be solved to find the time taken (t).
Please note that the time given by this equation will be the time taken by the water to reach the top, but it doesn't account for any air resistance or other factors that may affect the actual time taken in reality.
To determine the initial speed of the water and how long it takes for the water to reach the top of the fountain, we can use the principles of projectile motion. Here's how we can calculate it:
a. To find the initial speed of the water, we can use the equation for projectile motion:
v^2 = u^2 + 2a s
Where:
v = final velocity (which will be 0 at the top of the fountain)
u = initial velocity (what we want to find)
a = acceleration due to gravity (-9.8 m/s^2)
s = vertical displacement (height of the fountain, which is 560 ft or 170.688 m)
Since the final velocity is 0 at the top, the equation becomes:
0 = u^2 + 2a s
Rearranging the equation, we get:
u^2 = -2a s
Substituting the values into the equation:
u^2 = -2 * (-9.8 m/s^2) * 170.688 m
Simplifying the equation, we find:
u^2 = 3334.0416 m^2/s^2
Taking the square root of both sides, we get:
u ≈ 57.71 m/s
Therefore, the initial speed of the water is approximately 57.71 m/s.
b. To calculate the time it takes for the water to reach the top of the fountain, we can use the equation for vertical displacement in projectile motion:
s = ut + 0.5at^2
Where:
s = vertical displacement (height of the fountain, which is 560 ft or 170.688 m)
u = initial velocity (57.71 m/s, as calculated above)
a = acceleration due to gravity (-9.8 m/s^2)
t = time (what we want to find)
Plugging in the values, the equation becomes:
170.688 = (57.71)t + 0.5(-9.8)t^2
Re-arranging the equation, we get:
0.5(-9.8)t^2 + (57.71)t - 170.688 = 0
Now, we can solve this quadratic equation to find the value of t. Using a quadratic formula, we have:
t = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values, we get:
t = (-(57.71) ± √((57.71)^2 - 4 * 0.5 * (-9.8) * (-170.688))) / (2 * 0.5 * (-9.8))
After evaluating the expression, we find that:
t ≈ 8.29 s
Therefore, it takes approximately 8.29 seconds for the water to reach the top of the fountain.
a. v^2 = 2 g h = 2 * 32 * 560
b. average speed is half of the initial (zero at the top)
... time = distance / speed