Questions: A car is traveling at a speed of 104 km/h on the highway and has a small stone stuck between the treads of one of its tires. The tires have diameter 60.6 cm and are rolling without sliding or slipping. Part A What is the maximum speed of the stone as observed by a pedestrian standing on the side of the highway? Express your answer in kilometers per hour. Part B What is the minimum speed of the stone as observed by a pedestrian standing on the side of the highway? Express your answer in kilometers per hour.

A car is traveling at a speed of 104 km/h on the highway and has a small stone stuck between the treads of one of its tires. The tires have diameter 60.6 cm and are rolling without sliding or slipping.

Part A

What is the maximum speed of the stone as observed by a pedestrian standing on the side of the highway?
Express your answer in kilometers per hour.

Part B

What is the minimum speed of the stone as observed by a pedestrian standing on the side of the highway?
Express your answer in kilometers per hour.

Transcript text: A car is traveling at a speed of $104 \mathrm{~km} / \mathrm{h}$ on the highway and has a small stone stuck between the treads of one of its tires. The tires have diameter 60.6 cm and are rolling without sliding or slipping. Part A What is the maximum speed of the stone as observed by a pedestrian standing on the side of the highway? Express your answer in kilometers per hour. Part B What is the minimum speed of the stone as observed by a pedestrian standing on the side of the highway? Express your answer in kilometers per hour.

Solution

Solution Steps

Step 1: Understanding the Problem

The problem involves a car traveling at a speed of $104 \, \text{km/h}$ with a stone stuck in the tire tread. The tire has a diameter of $60.6 \, \text{cm}$. We need to find the maximum and minimum speeds of the stone as observed by a pedestrian.

Step 2: Determine the Tire's Angular Velocity

First, we convert the car's speed from kilometers per hour to meters per second: \[ 104 \, \text{km/h} = \frac{104 \times 1000}{3600} \, \text{m/s} = 28.8889 \, \text{m/s} \]

The radius of the tire is half of the diameter: \[ r = \frac{60.6 \, \text{cm}}{2} = 30.3 \, \text{cm} = 0.303 \, \text{m} \]

The angular velocity $\omega$ of the tire is given by: \[ \omega = \frac{v}{r} = \frac{28.8889}{0.303} \, \text{rad/s} = 95.3564 \, \text{rad/s} \]

Step 3: Calculate Maximum Speed of the Stone

The maximum speed of the stone occurs when it is at the top of the tire. At this point, the linear speed of the stone is the sum of the car's speed and the tangential speed due to the tire's rotation: \[ v_{\text{max}} = v + r\omega = 28.8889 + 0.303 \times 95.3564 = 57.7778 \, \text{m/s} \]

Convert this speed back to kilometers per hour: \[ v_{\text{max}} = 57.7778 \times \frac{3600}{1000} = 208 \, \text{km/h} \]

Step 4: Calculate Minimum Speed of the Stone

The minimum speed of the stone occurs when it is at the bottom of the tire. At this point, the linear speed of the stone is the car's speed minus the tangential speed due to the tire's rotation: \[ v_{\text{min}} = v - r\omega = 28.8889 - 0.303 \times 95.3564 = 0 \, \text{m/s} \]

Convert this speed back to kilometers per hour: \[ v_{\text{min}} = 0 \times \frac{3600}{1000} = 0 \, \text{km/h} \]