Now I am getting more confused. The answer just changed. Which one should I trust?

This is new answer

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Part $($a$)-$ Speed of the Center of Mass of Each Wheel

Rolling without slipping

Velocity at the contact point with the ground $0$

Velocity at the contact point with the plank v

Center of mass velocity

Rotational component R

Angular velocity $=$vR

Total velocity of the wheel's center

Translational velocity of the plank v

Adding the rotational component v$+$v$=2$v

Final Answer for Part $($a$)$ vwheel$=2$v

Explanation:

In a case where a plank bearing a certain speed $$v$,$ is rolling on wheels, these wheels will also be spinning. As the wheels neither slip on the plank nor on the ground the linear velocity at the point in contact with the ground is zero while that at the point in contact with the plank is equal to the speed of the plank. This means that the center of mass of the wheel oscillates within a speed that is compounded from the translational and rotational velocity of plank and wheels respectively. That is why the speed of the center of mass of each wheel is higher than the speed of the plank.

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