You and a friend are going to ride on a Ferris wheel. For a little extra fun, when your friend is at the very top of the Ferris wheel she's going to drop a tennis ball. How far around the Ferris wheel in radians from your friend should you sit so you can catch the ball?
Details and assumptions
The Ferris wheel radius is R=20 m.
The Ferris wheel goes around at a constant angular velocity ω=0.2 rad/s.
The acceleration of gravity is −9.8 m/s2.
Neglect air resistance.
i think its 5157.9442 radians
123665986956125361365362
To determine how far around the Ferris wheel in radians from your friend you should sit to catch the ball, we can follow the steps below:
Step 1: Determine the time it takes for the ball to fall
The time it takes for the ball to fall can be calculated using the formula for vertical displacement:
y = y0 + v0t + 0.5at^2
Where:
y = vertical displacement of the ball (in this case, it will be -R, since the ball is falling from the top of the Ferris wheel)
y0 = initial vertical position of the ball (in this case, it will be 0, as the ball is dropped from the top)
v0 = initial vertical velocity of the ball (in this case, it will be 0, as the ball is dropped without any initial velocity)
a = acceleration due to gravity (-9.8 m/s^2)
t = time
Rearranging the equation, we get:
t = √(2R / g)
where g is the acceleration due to gravity.
Step 2: Determine the angular displacement of the Ferris wheel during the time it takes for the ball to fall
The angular displacement can be calculated using the formula:
θ = ωt
where θ is the angular displacement, ω is the angular velocity of the Ferris wheel (0.2 rad/s), and t is the time calculated in Step 1.
Step 3: Calculate the distance in radians from your friend's position to where you should sit
Since your friend drops the ball at the very top of the Ferris wheel, her angular position at that moment is π radians or 180 degrees. To catch the ball, you need to sit at an angular position that is π radians + the angular displacement calculated in Step 2.
So, the distance in radians from your friend's position to where you should sit is:
π + θ
where π is approximately 3.14.
By plugging in the given values for R, ω, and g into the equations above, you can calculate the exact answer.