Question

At a county fair, a boy takes his teddy bear on the giant Ferris wheel. Unfortunately, at the top of the ride, he accidentally drops his stuffed buddy. The wheel has a diameter of \(12.0 \mathrm{~m},\) the bottom of the wheel is \(2.00 \mathrm{~m}\) above the ground and its rim is moving at a speed of \(1.00 \mathrm{~m} / \mathrm{s}\). How far from the base of the Ferris wheel will the teddy bear land?

Short answer

Answer: The teddy bear will land approximately 1.28 meters from the base of the Ferris wheel.

Step-by-step solution

Identify known quantities and what we need to find

We are given the following information: - Diameter of the Ferris wheel: 12 m - Bottom of the wheel is 2 m above the ground - Speed of Ferris wheel's rim: \(1.00 \mathrm{~m} / \mathrm{s}\) We need to find how far from the base of the Ferris wheel the teddy bear will land.

Calculate the height from which the teddy bear falls

We know the diameter of the Ferris wheel is 12 m, which means the radius is half of that: $$R = \frac{12}{2} = 6 \mathrm{~m}.$$ Given that the bottom of the wheel is 2 m above the ground, the total height, h, from which the teddy bear falls can be calculated as: $$h = R + 2 = 6 + 2 = 8 \mathrm{~m}.$$

Calculate the time it takes for the teddy bear to hit the ground

Now we'll use the kinematic equation to find the time, t, it takes for the teddy bear to hit the ground: $$h = \frac{1}{2}gt^2,$$ where g is the acceleration due to gravity (approximately \(9.81 \mathrm{~m/s^2}\)). Solve for t: $$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2(8)}{9.81}} \approx 1.28 \mathrm{~s}.$$

Calculate the horizontal distance teddy bear travels during the fall

Since the teddy bear will have the same horizontal velocity as the Ferris wheel's rim, we can use the equation: $$d = vt,$$ where d is the horizontal distance, v is the horizontal velocity, 1 m/s (same as the rim), and t is the time it takes to hit the ground (1.28 s). Calculate d: $$d = (1.00)(1.28) \approx 1.28 \mathrm{~m}.$$ So, the teddy bear will land approximately 1.28 meters from the base of the Ferris wheel.