To determine how far the second performer should stand, we need to find the horizontal distance the apple travels before it hits the ground. This involves breaking down the problem using the principles of projectile motion.

Key Information:

• Initial height: $h_0 = 33.0 \, \text{m}$
• Initial velocity: $v_0 = 23.0 \, \text{m/s}$
• Launch angle: $\theta = 20.0^\circ$
• Gravity: $g = 9.81 \, \text{m/s}^2$

Step 1: Decompose the initial velocity

We'll break the initial velocity into horizontal and vertical components:

• $v_{0x} = v_0 \cos(\theta)$ (horizontal component)
• $v_{0y} = v_0 \sin(\theta)$ (vertical component)

$v_{0x} = 23.0 \, \text{m/s} \times \cos(20.0^\circ)$ $v_{0y} = 23.0 \, \text{m/s} \times \sin(20.0^\circ)$

Step 2: Time of flight

We can calculate the time the apple spends in the air by solving for the time $t$ it takes for the apple to fall from 33.0 m, considering its initial upward velocity and downward acceleration due to gravity.

Using the vertical displacement formula:

$y(t) = v_{0y} t - \frac{1}{2} g t^2 + h_0$

Where $y(t) = 0$ (since it hits the ground), we solve for $t$.

Step 3: Horizontal distance

Once we know the time of flight $t$, the horizontal distance the apple travels is:

$x = v_{0x} \cdot t$

Let's calculate these steps now!It seems there's an issue with the calculation as the horizontal distance turned out to be negative. Let's correct the equation setup and reattempt.

In the quadratic equation for the time of flight, we should ensure the correct signs for gravity and vertical motion terms. I'll correct and calculate again.The horizontal distance the apple travels is approximately 76.01 meters. Therefore, the second performer should stand about 76 meters away from the end of the platform to catch the apple.

Would you like further details on any part of this solution, or do you have additional questions?

Here are some related questions:

1. How does air resistance affect projectile motion?
2. How can we determine the maximum height reached by the apple?
3. What if the performer throws the apple at a different angle? How would the distance change?
4. What happens if the initial height changes?
5. How does changing gravity (e.g., on another planet) affect the horizontal distance?

Tip: In projectile motion, the horizontal and vertical motions are independent of each other, with the only link being the time of flight.