Fairgoers ride a Ferris wheel with a radius of 5.00 {\rm m} . The wheel completes one revolution every 31.5 s

What is the average speed of a rider on this Ferris wheel?

If a rider accidentally drops a stuffed animal at the top of the wheel, where does it land relative to the base of the ride? (Note: The bottom of the wheel is 1.75 {\rm m} above the ground.)

To determine the average speed of a rider on the Ferris wheel, we need to use the formula for average speed, which is calculated by dividing the total distance traveled by the total time taken.

First, we need to find the total distance traveled by the rider. Since the Ferris wheel is a circle, the distance traveled by the rider during one complete revolution is equal to the circumference of the circle.

The circumference of a circle can be calculated using the formula:
C = 2πr

Where C is the circumference and r is the radius of the circle.

In this case, the radius of the Ferris wheel is given as 5.00 m. Plugging this value into the formula, we get:
C = 2π * 5.00 = 10π

Next, we need to find the total time taken for one complete revolution. The problem states that the wheel takes 31.5 seconds to complete one revolution.

Now we can calculate the average speed using the formula:
Average Speed = Total Distance / Total Time

Substituting the values we found, we get:
Average Speed = 10π / 31.5

Calculating the approximate value, we get:
Average Speed ≈ 0.318 m/s (rounded to 3 decimal places)

Therefore, the average speed of a rider on this Ferris wheel is approximately 0.318 m/s.

Now, let's determine where the stuffed animal dropped from the top of the wheel would land relative to the base of the ride.

Since the bottom of the wheel is 1.75 m above the ground, and assuming there is no air resistance affecting the motion of the stuffed animal, it will follow a parabolic trajectory rather than simply dropping straight down.

At the top of the wheel, the stuffed animal has an initial velocity of zero, and it will experience a downward acceleration due to gravity.

Using laws of motion, we can calculate the horizontal distance traveled by the stuffed animal. Since there is no horizontal force acting on it, the horizontal velocity remains constant.

The time it takes for the stuffed animal to reach the ground can be found using the equation:
1.75 = (1/2) * g * t^2

Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time taken to reach the ground.

Solving this equation gives us:
t^2 = (2 * 1.75) / 9.8
t^2 ≈ 0.357

Taking the square root, we find:
t ≈ 0.598 s (rounded to 3 decimal places)

Now we can calculate the horizontal distance traveled using the equation:
Horizontal Distance = Horizontal Velocity * Time

Since the initial horizontal velocity is zero, the horizontal distance traveled will also be zero.

Therefore, the stuffed animal will land at the base of the ride directly beneath the point it was dropped from, relative to the base of the ride.

To calculate the average speed of a rider on the Ferris wheel, we can use the formula for linear motion:

Average Speed = Total Distance / Total Time

1. The total distance covered by the rider is the circumference of the Ferris wheel, which is given by the formula:

Circumference = 2 * π * radius

Plugging in the values, we get:

Circumference = 2 * 3.14 * 5.00 m = 31.40 m

2. The total time taken for one revolution of the Ferris wheel is 31.5 seconds.

Now we can calculate the average speed:

Average Speed = Circumference / Total Time
Average Speed = 31.40 m / 31.5 s = 0.995 m/s

Therefore, the average speed of a rider on this Ferris wheel is approximately 0.995 m/s.

3. If a rider accidentally drops a stuffed animal at the top of the wheel, it will fall from the top to the bottom due to gravity. The distance from the top of the Ferris wheel to the ground is the sum of the radius of the wheel and the height of the base.

Height from top to base = radius + height of base
Height from top to base = 5.00 m + 1.75 m = 6.75 m

So, the stuffed animal will land approximately 6.75 meters below the top of the Ferris wheel, relative to the base of the ride.

The angular speed is

w = (2 pi radians)/31.5 s = 0.2 rad/s

The speed of a rider remains V = R*w at all times

I cannot interpret your {\rm symbol.

If a stuffed animal is droppped at the top of the whhel, it starts with a horizonal velocity V = R*w.

Proceed in the usual way (Newton's laws) to get the time to fall and the place where it lands after that