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 15.3 m, the bottom of the wheel is 3.37 m above the ground and its rim is moving at a speed of 1.93 m/s. How far from the base of the Ferris wheel will the teddy bear land?

To determine how far from the base of the Ferris wheel the teddy bear will land, we need to calculate the horizontal distance traveled by the bear from the moment it was dropped until it lands.

First, we can find the time it takes for the bear to fall from the top of the Ferris wheel to the ground. We will use the equation for time of free fall:

t = sqrt((2h) / g),

where h is the height and g is the acceleration due to gravity.

Given that the height of the Ferris wheel is the sum of the wheel's radius and its height above the ground, we can calculate h:

h = 3.37 m + (15.3 m / 2) = 11.02 m.

The acceleration due to gravity, g, is approximately 9.8 m/s^2.

Plugging these values into the equation, we get:

t = sqrt((2 * 11.02 m) / 9.8 m/s^2) ≈ 1.48 s.

Now, we can calculate the horizontal distance traveled by the bear during this time. The horizontal distance is given by the formula:

d = v * t,

where v is the horizontal velocity of the point on the rim of the Ferris wheel.

Given that v = 1.93 m/s, we can substitute the values:

d = 1.93 m/s * 1.48 s ≈ 2.86 m.

Therefore, the teddy bear will land approximately 2.86 meters from the base of the Ferris wheel.

To find out how far from the base of the Ferris wheel the teddy bear will land, we need to calculate the horizontal distance traveled by the teddy bear while it is falling.

First, we can calculate the time it takes for the teddy bear to fall from the top of the Ferris wheel to the ground. We can use the formula for free fall:

h = 1/2 * g * t^2

Where h is the height from the top of the Ferris wheel to the ground, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time in seconds.

Since the height from the top of the Ferris wheel to the ground is the sum of the height of the Ferris wheel (3.37 m) and the diameter of the Ferris wheel (15.3 m), we have:

h = 3.37 m + 15.3 m = 18.67 m

Plugging in the values, we can solve for t:

18.67 = 1/2 * 9.8 * t^2
37.34 = 9.8 * t^2
t^2 = 37.34 / 9.8
t^2 = 3.81
t ≈ √3.81
t ≈ 1.95 s

Now that we have the time, we can calculate the horizontal distance traveled by the teddy bear while falling using:

d = v * t

Where v is the horizontal speed of the Ferris wheel rim (1.93 m/s) and t is the time in seconds.

Plugging in the values, we have:

d = 1.93 m/s * 1.95 s
d ≈ 3.76 m

Therefore, the teddy bear will land approximately 3.76 meters from the base of the Ferris wheel.