Question

As a car travels along a road, the speed of the tops of its wheels is \(46 \mathrm{~m} / \mathrm{s}\). What is the speed of the car and its occupants?

Short answer

The speed of the car and its occupants is 23 m/s.

Step-by-step solution

Understanding the Relationship

The speed of the top of the wheel is twice the speed of the car itself. This is because the wheel rolls on the road, and its top moves faster than the center of the wheel, which moves at the speed of the car.

Setting Up the Equation

Let the speed of the car be denoted by \( v \). Thus, the speed at the top of the wheel will be \( 2v \). We are given that the speed of the top of the wheel is \( 46 \, \text{m/s} \), so we can set the equation \( 2v = 46 \).

Solving for the Car's Speed

Solve the equation \( 2v = 46 \) for \( v \) by dividing both sides by 2. This gives us \( v = \frac{46}{2} = 23 \). So, the speed of the car is \( 23 \, \text{m/s} \).

Key concepts

Kinematics

Kinematics is a fundamental branch of physics that deals with the motion of objects. It describes how objects move, but not why they move. This distinction is important, as kinematics focuses on parameters like velocity, acceleration, and displacement, without explaining the forces involved.

• Displacement: This refers to the change in position of an object. It's a vector quantity, meaning it has both magnitude and direction.
• Velocity: The speed of an object in a particular direction. It's also a vector quantity. In our exercise, the speed of the car and the speed at the top of the wheel represent velocities at different points.
• Acceleration: The rate of change of velocity with time. Knowing the acceleration can help us predict future motion of an object.

Kinematics allows us to develop equations that help solve various problems related to motion—like predicting how far an object will travel or how long it will take to reach a certain speed. In the context of our car exercise, understanding kinematics helps us establish the relation between different parts of a moving object, such as the car and its wheels.

Wheel Dynamics

Wheel dynamics is critical to understanding how wheels interact with cars and the road. The main concept is that a wheel rotates about its axis while simultaneously moving linearly with the car. The velocity of different points on a wheel is not uniform due to this rotational movement. A key point in wheel dynamics relevant to our problem is that the top of a wheel moves at a velocity that's twice the car's speed. This is because the wheel's center moves at the car’s speed, and due to rotation, the top portion effectively adds the wheel's speed to this, resulting in twice the speed. To break it down:

• Center of the Wheel: Moves with the linear speed of the car.
• Top of the Wheel: Its speed is the combination of linear speed from the car and the rotational speed, thus reaching twice the car's speed.
• Bottom of the Wheel: Momentarily at rest relative to the ground as it "kisses" the road.

Understanding these dynamics allows us to set up equations that relate the speed of the car to the observed speed of the wheel’s top, such as the given equation in the original step-by-step solution.

Speed Calculation

Speed calculation involves determining the rate at which an object covers distance. It's a fundamental concept in motion analysis, aiding in understanding how fast an object moves from one point to another. The speed is simply the distance traveled over time taken, often simplified in problems where other factors, like rotation in wheel dynamics, are at play.To solve the exercise, we first recognize that the speed at the top of the wheel is double the car’s speed because of its rotational motion. Thus, by setting up the equation as given in the solution, we start:We have:\[2v = 46 \]To find the car's speed \(v\), we divide both sides by 2:\[v = \frac{46}{2} = 23 \, \text{m/s}\]This equation captures the essence of linking rotational and translational motion. Mastering speed calculations provides insight into the broader implications of movement and assists with solving complex problems involving multiple functional relationships between objects in motion.