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.
To determine how far around the Ferris wheel in radians you should sit in order to catch the ball, we'll need to consider the motion of the ball as well as the motion of the Ferris wheel.
First, let's calculate the time it takes for the ball to fall from the top of the Ferris wheel to the ground. We can use the equation for the vertical position of an object in free fall:
h = h0 + v0t - (1/2)gt^2
Where:
- h is the vertical position of the ball (measured from the top of the Ferris wheel)
- h0 is the initial vertical position of the ball (equal to the height of the Ferris wheel, which is R = 20 m)
- v0 is the initial vertical velocity of the ball (equal to 0 since the ball is dropped)
- g is the acceleration due to gravity (-9.8 m/s^2)
- t is the time it takes for the ball to fall
Since the initial vertical velocity is 0, the equation simplifies to:
h = h0 - (1/2)gt^2
At the top of the Ferris wheel, h is 0 (since it's at the highest point) and h0 is R. Therefore, we can solve for t:
0 = R - (1/2)gt^2
Simplifying the equation:
(1/2)gt^2 = R
gt^2 = 2R
t^2 = (2R)/g
t = sqrt((2R)/g)
Now that we have the time it takes for the ball to fall, we can calculate the angle through which the Ferris wheel rotates during that time. The angular displacement is given by:
θ = ωt
Where:
- θ is the angular displacement in radians
- ω is the angular velocity of the Ferris wheel (0.2 rad/s)
- t is the time calculated previously
Substituting the values:
θ = (0.2 rad/s)(sqrt((2R)/g))
Now, we can calculate the distance around the Ferris wheel in radians from your friend at the top to the position where you should sit to catch the ball. Since your friend is at the very top, her angular displacement is 0 radians. Therefore, the distance in radians you should sit is:
π - θ
Finally, substitute the given values of R, ω, and g into the equation to find the desired position:
θ = (0.2 rad/s)(sqrt((2(20 m))/(-9.8 m/s^2))) = 0.288 rad
Therefore, you should sit approximately π - 0.288 radians around the Ferris wheel from your friend to catch the ball.