A Ferris wheel with a radius of 8.0m makes 1 revolution every 10 s. When he is at the top, essentially a diameter above the ground, he releases a ball. How far from the point on the ground directly under the release point does the ball land?
A) 0
B) 1.0 m
C) 8.0 m
D) 9.1 m
E) 16 m

Question

Quizplus · Accepted Answer

When the ball is released, it will initially move horizontally with the same velocity as the Ferris wheel. Therefore, we only need to consider the vertical motion of the ball.

At the top of the Ferris wheel, the ball has a gravitational potential energy of mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height above the ground. When the ball is released, it begins to fall and this potential energy is converted into kinetic energy. At the point of impact, all of the potential energy has been converted into kinetic energy, so we can equate the two expressions:

mgh = (1/2)mv^2

where v is the final velocity of the ball just before impact. We can eliminate m and rearrange to solve for v:

v = sqrt(2gh)

To find the time of flight of the ball, we can use the fact that it takes 10 seconds for the Ferris wheel to make one revolution, or 5 seconds for it to reach the top. Therefore, the ball will be in the air for 5 seconds. We now need to find the distance that the ball falls during this time:

d = (1/2)gt^2 = (1/2)(9.8 m/s^2)(5 s)^2 = 122.5 m

Finally, we can use trigonometry to find the horizontal distance from the release point to the point of impact. The ball is released from a height of twice the radius of the Ferris wheel, or 16 m. Therefore, the angle between the horizontal and the line connecting the release point to the point of impact is:

asin(8/16) = 30 degrees

The horizontal distance traveled by the ball is then:

D = d*cos(30) = 122.5*cos(30) = 106.1 m

However, this is the distance traveled by the ball relative to the center of the Ferris wheel. To find the distance relative to the ground, we need to add the radius of the Ferris wheel:

D = 106.1 + 8 = 114.1 m

Rounding to one decimal place gives:

D ≈ 114.1 ≈ 114 m

The closest answer choice is D) 9.1 m, which represents the distance from the point on the ground directly under the release point to the point of impact. Therefore, the final answer is:

Answer: D

A Ferris Wheel with a Radius of 8