A circus performer throws an apple toward a hoop held by a performer on a platform (see figure below). The thrower aims for the hoop and throws with a speed of 30 m/s. At the exact moment the thrower releases the apple, the other performer drops the hoop. The hoop falls straight down. (Assume

d = 19 m and h = 55 m.
Neglect the height at which the apple is thrown.)
(a) At what height above the ground does the apple go through the hoop?

m

(b) If the performer on the platform did not drop the hoop, would the apple pass through it?
Yes
No

d, h ?? What are these?

To find the height at which the apple goes through the hoop, we can analyze the projectile motion of the apple.

(a) First, let's split the motion into horizontal and vertical components. The horizontal velocity of the apple remains constant throughout the motion. Since the hoop is dropped vertically, its horizontal velocity does not affect the motion vertically.

The vertical motion of the apple can be analyzed using the equation for vertical displacement in projectile motion:

y = y0 + v0y * t + (1/2) * a * t^2

where y is the vertical displacement, y0 is the initial vertical position, v0y is the initial vertical velocity, a is the acceleration due to gravity (-9.8 m/s^2), and t is the time.

In this case, the initial vertical position (y0) is 0 (as the apple starts from the ground), the initial vertical velocity (v0y) is 0 (since the height at which the apple is thrown is neglected), and the vertical displacement (y) is the height at which the apple goes through the hoop.

We can rearrange the equation to solve for t:

y = (1/2) * a * t^2

Solving for t:

t = sqrt(2 * y / a)

Substituting the given values:

t = sqrt(2 * 19 m / 9.8 m/s^2) = 2.19 s

Now, we can find the height at which the apple goes through the hoop by substituting the value of t into the equation for vertical displacement:

y = (1/2) * a * t^2 = (1/2) * 9.8 m/s^2 * (2.19 s)^2 = 23.7 m

Therefore, the apple goes through the hoop at a height of 23.7 meters above the ground.

(b) Since the height at which the apple is thrown is neglected and the apple goes through the hoop at a height of 23.7 meters, which is greater than the initial height, the apple would not pass through the hoop if the performer on the platform did not drop it. Therefore, the answer is "No".