a question about dynamics?

No. The velocity of the wheel is changing as the car moves (because velocity indicates direction). So as the car moves forward, the velocity of the surface of the wheel is continuously changing.

The velocity is a vector quantity and two velocities will be equal only when both magnitude and direction of the two vectors are same.

The top surface of the wheel is moving in the direction of the motion of the car and also it has a rotational speed in the direction of the motion of the car.

Therefore the two velocities are added together to have resultant velocity with a magnitude twice that of car’s velocity.

Let the car be moving with a speed of 20m per second (72 km/ hour).

Let us assume that the wheels are rotating in anti- clockwise direction.

Each point on the circumference of the wheel has two velocities, one 20m/s in the direction of motion of the car and another 20m/s along the tangent to the path of the curve along which the point moves.

One has to find the resultant velocity of these two velocities.

The path of a point on the circumference of the wheel is a cycloid.

The direction of velocity is along the direction of the tangent drawn at the point on the cycloid curve.

Consider a point P on the wheel.

When it is touching the ground its speed is zero. At that instant it is neither moving up nor moving down, and it is not moving front or back. The car’s velocity 20m/s is in the opposite direction of the point p’s velocity in the backward direction and the net velocity is zero.

When the wheel has turned quarter revolution, the speed of the point P is 20m/s in the upward direction and 20m/s in the forward direction. The resultant speed is 28.3 m/s in a direction inclined at angle 45 degrees to the vertical (clockwise)

When the wheel has turned half revolution, the speed of the point P is 20m/s in the direction of the motion of the car and the car’s speed of 20m/s. Thus it has a speed of 40m/s in the direction of the car.

When the wheel has turned 3/4th revolution, the speed of the point P is 20m/s in the downward direction and 20m/s in the forward direction. The resultant speed is 28.3 m/s in a direction inclined at angle135 degrees to the vertical (clockwise)

When the wheel has turned one full revolution, the speed of the point P is again zero making the point instantaneously at rest.

Real simple. Is the top surface moving with respect to the moving car? Yes. If you don't believe it try riding a bike and feel the top surface of the front wheel. If the car's velocity is V, the bottom surface has zero velocity with respect to the pavement thus -V with respect to the car; the wheel center has zero velocity with respect to the car, the top has a velocity +V with respect to the car.

Actually, this question almost got me. The bottom surface of the wheel is rolling across the surface at the same speed as the vehicle. The top cannot be moving any faster than the rest of the wheel, therefore, your answer is yes.

The top of the wheel and the vehicle are traveling at the same velocity.

Hmm, good question. Let's consider the bottom of the wheel. Since the road moves back at the same rate that the axle, and car, moves forward, then the bottom moves at the same speed as the car, right?

Intuition says that it moves twice as fast. Take the velocity vector of the axle (=speed of car) and add the r*(angular velocity).

It's still a good question, I just don't have a good answer readily available.

no the velocity of the car is equal to half the velocity of the top surface of th wheel if the wheel is rolling

no, its twice as fast. just before it gets to the top, it is moving up away from the road surface and forward, in the same direction as the car. as it crosses over the top it begins moving downward toward the road surface and still forward. when it gets to the bottom, it momentarily stops- not moving forward and changing direction from down to up.

This can be proven by writing the equation of a circle and integrating to find the maximum point- which turns out to be the top of the circle or top surface of the wheel.

no...just no.