Let's break down the problem to determine how far the second performer should stand to catch the apple.

Given Information:

• Height of the platform: $h = 40.0 \, \text{m}$
• Initial velocity: $v_0 = 17.0 \, \text{m/s}$
• Angle of projection: $\theta = 30.0^\circ$ (measured downward from the horizontal)
• Acceleration due to gravity: $g = 9.81 \, \text{m/s}^2$

We need to find how far from the platform (horizontally) the second performer should stand, denoted as $x$.

Step 1: Break the initial velocity into components

We decompose the initial velocity $v_0$ into horizontal and vertical components:

• Horizontal component: $v_{0x} = v_0 \cos(\theta)$ $v_{0x} = 17.0 \times \cos(30^\circ) = 17.0 \times 0.866 = 14.72 \, \text{m/s}$

• Vertical component: Since the apple is thrown downward, the vertical component is: $v_{0y} = -v_0 \sin(\theta)$ $v_{0y} = -17.0 \times \sin(30^\circ) = -17.0 \times 0.5 = -8.5 \, \text{m/s}$

Step 2: Time of flight

We use the vertical motion equation to determine the time the apple is in the air: $y = y_0 + v_{0y} t + \frac{1}{2} g t^2$ Where:

• $y_0 = 40.0 \, \text{m}$
• $y = 0$ (ground level)
• $v_{0y} = -8.5 \, \text{m/s}$
• $g = 9.81 \, \text{m/s}^2$

Substitute values: $0 = 40.0 - 8.5 t + \frac{1}{2} \times 9.81 \, t^2$ Simplify: $4.905 t^2 - 8.5 t - 40 = 0$ This is a quadratic equation. Solving it using the quadratic formula: $t = \frac{-(-8.5) \pm \sqrt{(-8.5)^2 - 4 \times 4.905 \times (-40)}}{2 \times 4.905}$ $t = \frac{8.5 \pm \sqrt{72.25 + 784.8}}{9.81}$ $t = \frac{8.5 \pm \sqrt{857.05}}{9.81}$ $t = \frac{8.5 \pm 29.28}{9.81}$ We take the positive root since time must be positive: $t = \frac{8.5 + 29.28}{9.81} = \frac{37.78}{9.81} \approx 3.85 \, \text{seconds}$

Step 3: Horizontal distance traveled

Now that we have the time, the horizontal distance $x$ can be calculated using: $x = v_{0x} t$ $x = 14.72 \times 3.85 \approx 56.7 \, \text{m}$

Final Answer:

The second performer should stand 56.7 meters from the end of the platform to catch the apple.

Would you like more details or have any questions?

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Here are five related questions you might find useful:

1. What happens if the apple is thrown at a higher angle?
2. How would the horizontal distance change if the initial speed increased?
3. What if the second performer stands on an elevated platform? How would that affect the calculation?
4. What is the role of air resistance in this calculation?
5. How would the result change if the apple was thrown at an upward angle?

Tip: When dealing with projectiles, always decompose the velocity into horizontal and vertical components, as it simplifies calculations involving motion.