Understanding Wheel Speed at Various Points for a Car Moving at 100 mph

Introduction

The concept of wheel speed when a car is in motion can sometimes be confusing, especially when dealing with specific points on the wheel such as the highest and lowest points on the vertical diameter. This article delves into the physics behind these scenarios, providing clarity on the speed of each part of the wheel as it rotates while the car is moving at 100 mph.

Instantaneous Speed at an Instant

It is important to clarify the notion of speed at a single, instantaneous moment. Speed, defined as distance over time, implies a process that occurs over a period, not a single point in time. Therefore, at an instant in time, an object must have a non-zero distance and a non-zero time, neither of which can be zero. This makes the speed undefined at an instant, as dividing by zero is not mathematically valid.

Car Velocity and Wheel Angular Velocity

A car moving at 100 mph has a constant forward velocity. However, the wheels of the car are also rotating at a certain angular velocity. Given that the car is moving at 100 mph and each wheel has a radius of R, the angular velocity ω of the wheel is calculated as:

ω v / R

Where:

v 100 mph ω angular velocity of the wheel

Each point on the circumference of the wheel is moving at a speed of:

v Rω

relative to the center of the wheel. However, these points are moving in a direction that is 90 degrees ahead of their radial position, meaning they have different directions of motion.

Speed at the Top and Bottom of the Wheel

Let's consider the top and bottom points of the wheel's vertical diameter to understand the relative movement better.

Top of the Wheel: The top of the wheel is moving in the same direction as the car's forward velocity. Therefore, to a stationary observer on the road, the top of the wheel will be moving at:

v_top v 100 mph

However, considering the angular speed, the top of the wheel will be:

v_top 2v 200 mph

relative to the road, due to the additional circular motion of the wheel.

Bottom of the Wheel: The bottom of the wheel, on the other hand, moves in the opposite direction of the car's forward velocity. Therefore, to a stationary observer on the road, the bottom of the wheel will be moving at:

v_bottom v 100 mph

backward with respect to the center of the wheel. Therefore, the speed of the bottom of the wheel relative to the road is:

v_bottom 0 mph

This is because the point of contact with the ground is momentarily at rest relative to the road surface.

Relative Perspective of the Passenger and Observer

The speed perceptions of a passenger in the car and an observer at the side of the road differ. For a passenger inside the car:

The top of the wheel is moving at 100 mph towards the front of the car. The bottom of the wheel is moving at 100 mph towards the rear of the car.

For an observer at the side of the road:

The top of the wheel moves at 200 mph towards the front of the car. The bottom of the wheel remains stationary as it is in contact with the ground.

Visualizing with a Lever

To further illustrate, imagine a car's wheel replaced by a vertical lever with the center of the wheel (axle) being lifted above the ground. The lever's top end moves at twice the speed of the center (since it is twice the radius) while the bottom end remains stationary.

In this scenario, the top end of the lever, representing the top of the wheel, moves at 200 mph. The bottom end, representing the bottom of the wheel, remains stationary when in contact with the ground.

This setup aligns with the actual physical behavior of a car's wheels, where the point of contact with the ground does not move relative to the ground.

Conclusion

The motion of a car's wheels involves both linear and angular components, leading to complex relative speeds at different points on the wheel. Understanding these concepts helps in visualizing and appreciating the physics behind vehicle motion.