At a county fair, a boy and two friends bring their teddy bears on the giant Ferris wheel. The wheel has a diameter of 14.3 m, the bottom of the wheel is 1.9 m above the ground and its rim is moving at a speed of 1.0 m/s. The boys are seated in positions 45° from each other. When the wheel brings the second boy to the maximum height, they all drop their stuffed animals. How far apart will the three teddy bears land? (Assume that the boy on his way down drops bear 1, and the boy on his way up drops bear 3.)

distance between bears 1 and 2?
distance between bears 2 and 3?

To solve this question, we need to consider the motion of the Ferris wheel and the motion of the teddy bears after they are dropped.

First, let's calculate the height of the Ferris wheel at the maximum height. Since the bottom of the wheel is 1.9 m above the ground, the maximum height of the Ferris wheel can be calculated by adding the radius of the wheel (half of the diameter) to the 1.9 m. The radius of the wheel is 14.3 m divided by 2, which is 7.15 m. Therefore, the maximum height of the Ferris wheel is 7.15 m + 1.9 m = 9.05 m.

Now, let's calculate the time it takes for the teddy bears to fall from the maximum height to the ground. We can use the equation of motion, h = 1/2 * g * t^2, where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time. Solving for t, we get t = sqrt(2h/g). Plugging in the values, t = sqrt(2 * 9.05 m / 9.8 m/s^2) = 1.37 s.

Since the teddy bears are dropped when the second boy is at the maximum height, we can assume that the time for the first teddy bear to fall is t, and the time for the third teddy bear to fall is also t.

Next, let's calculate the distance traveled by the first teddy bear, from the point it is dropped to the ground. We can use the equation of motion, d = v*t + 1/2 * g * t^2, where d is the distance, v is the initial velocity (0 m/s since the bear is dropped), t is the time, and g is the acceleration due to gravity. Plugging in the values, d = 0 + 1/2 * 9.8 m/s^2 * (1.37 s)^2 = 9.07 m.

Now, let's calculate the distance between the first and second teddy bears. Since the boys are seated 45° from each other, the angle between the two teddy bears will also be 45°. We can calculate this distance using the formula for the circumference of a circle, C = 2πr, where C is the circumference, and r is the radius. The radius of the ferris wheel is 7.15 m, so the circumference is 2 * π * 7.15 m = 14.26 m. Since there are 360° in a circle, 45° is 45/360 = 1/8 of the circumference. Therefore, the distance between the first and second teddy bears is 1/8 * 14.26 m = 1.78 m.

Finally, let's calculate the distance between the second and third teddy bears. Since the Ferris wheel is moving at a constant speed of 1.0 m/s, the distance traveled by the second boy from the maximum height to the point where the third boy is dropped is equal to the circumference of the Ferris wheel. Therefore, the distance between the second and third teddy bears is also 14.26 m.

So, the distance between the first and second teddy bears is 1.78 m, and the distance between the second and third teddy bears is 14.26 m.