Suppose the water near the top of Niagara Falls has a horizontal speed of 2.7 m/s just before it cascades over the edge of the falls. At what vertical distance below the edge does the velocity vector of the water point downward at a 72° angle below the horizontal?
The water fows at that angle when the ratio Vy/Vx = tan 72 = 3.08
Vx remains 2.7 m/s
Vy = sqrt(2Y/g) = 3.08*2.7 = 8.3 m/s
Calculate Y (the distance the water has fallen) using that value of Vy.
To solve this problem, we need to use trigonometry and the concept of projectile motion. Let's break down the steps:
Step 1: Resolve the velocity into horizontal and vertical components.
Given that the water's horizontal speed is 2.7 m/s, we can find the horizontal component (Vx) as:
Vx = velocity * cos(angle)
Vx = 2.7 m/s * cos(72°)
Step 2: Calculate the vertical component of the velocity.
The vertical component (Vy) of the velocity can be found using the relationship:
Vy = velocity * sin(angle)
Vy = 2.7 m/s * sin(72°)
Step 3: Determine the time it takes for the water to reach the vertical distance.
To find the time it takes for the water to reach a certain vertical distance, we need to use the equation:
Vertical distance = Vy * time - (1/2) * g * time^2
Since the water is already at the top of Niagara Falls, the initial vertical distance is zero. Therefore, the equation becomes:
0 = Vy * time - (1/2) * g * time^2
Step 4: Solve for time.
Rearrange the equation to solve for time:
(1/2) * g * time^2 = Vy * time
(1/2) * g * time = Vy
time = (2 * Vy) / g
Step 5: Substitute in the values and calculate the time.
Substitute the values of Vy and g into the equation:
time = (2 * Vy) / g
time = (2 * 2.7 m/s * sin(72°)) / 9.8 m/s^2
Step 6: Calculate the vertical distance below the edge.
Now that we have the time it takes for the water to reach a certain vertical distance, we can calculate the vertical distance below the edge using the equation:
Vertical distance = Vy * time
Substitute the values of Vy and time into the equation:
Vertical distance = 2.7 m/s * sin(72°) * (2 * 2.7 m/s * sin(72°)) / 9.8 m/s^2
Simplify the equation to find the vertical distance below the edge of Niagara Falls.