Question

Fairgoers ride a Ferris wheel with a radius of \(5.00 \mathrm{~m}\), as shown in Figure 4.40. The wheel completes one revolution every \(32.0 \mathrm{~s}\). (a) What is the average speed of a rider on this Ferris wheel? (b) 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 \(1.75 \mathrm{~m}\) above the ground.)

Short answer

(a) Average speed: \(0.981 \mathrm{~m/s}\). (b) Object lands \(1.51 \mathrm{~m}\) from base.

Step-by-step solution

Calculate the Circumference of the Ferris Wheel

The circumference of a circle, which is the path the rider travels, is given by the formula \( C = 2\pi r \), where \( r \) is the radius of the Ferris wheel. Here, \( r = 5.00 \mathrm{~m} \). Thus, the circumference \( C = 2 \pi \times 5.00 \mathrm{~m} = 10\pi \mathrm{~m} \).

Determine the Average Speed

The average speed is calculated as the total distance traveled divided by the time taken. As the wheel completes one revolution every \( 32.0 \mathrm{~s} \), the average speed \( v = \frac{10\pi \mathrm{~m}}{32.0 \mathrm{~s}} \approx 0.981 \mathrm{~m/s} \).

Consider the Path of the Dropped Object

When the stuffed animal is dropped from the top of the Ferris wheel, it will follow a projectile motion path due to gravity. It is released at a height of \( h = 2r + 1.75 \mathrm{~m} \), as the top of the Ferris wheel is \( 2r = 10.0 \mathrm{~m} \) above the bottom. So, the total height is \( 10.0 + 1.75 = 11.75 \mathrm{~m} \).

Calculate the Horizontal Distance Traveled

The time \( t \) it takes for the stuffed animal to fall can be calculated using the formula for free fall \( h = \frac{1}{2}gt^2 \), where \( g = 9.81 \mathrm{~m/s^2} \). Solving for \( t \), we have \( 11.75 = \frac{1}{2} \times 9.81 \times t^2 \), which gives \( t \approx 1.54 \mathrm{~s} \).

Finalize Where the Object Lands Horizontally

Calculate where the object lands horizontally with respect to the base by considering the horizontal speed of the ferris wheel at the top. Using \( v \approx 0.981 \mathrm{~m/s} \), the horizontal distance \( d = v \times t = 0.981 \times 1.54 \approx 1.51 \mathrm{~m} \). Thus, the stuffed animal lands approximately \( 1.51 \mathrm{~m} \) horizontally from the point directly below where it was dropped.

Key concepts

Projectile Motion

When an object is thrown or dropped in the air, it moves in a curved path called projectile motion. In the case of the stuffed animal falling from the Ferris wheel, gravitational force affects it by pulling it downward. Meanwhile, the motion of the Ferris wheel gives it an initial horizontal velocity. Together, these forces create a parabolic trajectory, a hallmark of projectile motion.

Key points to remember about projectile motion:

• The only force acting on the object is gravity, assuming no air resistance.
• The initial velocity can have both horizontal and vertical components.
• The path of the motion is typically a parabola.
• In any projectile motion, the horizontal motion and vertical motion are independent of each other.

Understanding these factors helps us predict where the object, in this case, a stuffed animal, will land.
The horizontal distance traveled by the product can be calculated using the object’s horizontal speed and the time it takes to fall. This predicts where the dropped object will land relative to its original point of release.

Average Speed

Average speed is a basic concept in circular motion and any type of motion. It defines how fast an object travels along its path. In the scenario with the Ferris wheel, we're interested in how fast a passenger moves as the wheel spins around.

To determine average speed, you'll need:

• The total distance traveled – for a circular path, this equates to the circumference of the wheel.
• The time taken to complete one full revolution.

This forms the equation for average speed:\[v = \frac{\text{total distance}}{\text{total time}}\]For the Ferris wheel, the calculation was:\[v \approx \frac{10\pi \text{ m}}{32.0 \text{ s}} \approx 0.981 \text{ m/s}\]
This means that at any given point, a rider moves at an average speed of approximately 0.981 meters per second along the circular path of the Ferris wheel. Understanding average speed helps to comprehend the basics of motion, whether in circular paths or straight lines.

Free Fall

Free fall describes the motion of an object under the influence of gravity alone. When the stuffed animal is dropped from the top of the Ferris wheel, it experiences free fall, as gravity is the sole force acting on it.
Remarkable features of free fall include:

• Only gravity affects the object's motion.
• It accelerates downwards at approximately \( 9.81 \text{ m/s}^2 \), the acceleration due to Earth's gravity.
• Time taken to fall can be determined using the free fall formula:\[ h = \frac{1}{2} g t^2 \]where \( h \) is the height and \( g \) is gravitational acceleration.

In our situation, for a height of 11.75 meters, it took roughly 1.54 seconds for the stuffed animal to reach the ground.
This understanding of time and distance in free fall is essential for solving problems where objects are dropped or fall freely under gravity's influence.