When a wheel is rolling along the ground without slipping, different points on the wheel have different velocities. Here are the velocities of the top, bottom, front, and back points on the wheel:
Top Point: The top point of the wheel is moving in the same direction as the center point (east in this case) at the same speed. So, its velocity is 10 m/s [E].
Bottom Point: The bottom point of the wheel is also moving in the same direction as the center point at the same speed. So, its velocity is 10 m/s [E].
Front Point: The front point of the wheel is moving forward (east) with the velocity of the center, which is 10 m/s [E]. In addition, it's also moving due to the rotation of the wheel. Since it's on the circumference of the wheel, it moves a distance equal to the wheel's circumference in the same time it takes for one full rotation. The circumference of the wheel is 2π times its radius.
v_front = Velocity due to translation (10 m/s [E]) + Velocity due to rotation
v_front = 10 m/s [E] + (2π * 0.5 m * 10 m/s) [N]
Now, calculate the value:
v_front = 10 m/s [E] + 31.42 m/s [N]
You can calculate the magnitude and direction of this vector using the Pythagorean theorem and trigonometry. The magnitude is approximately 32.2 m/s, and the direction is approximately 17 degrees north of east.
Back Point: The back point of the wheel is moving backward (west) with the velocity of the center, which is 10 m/s [E]. It's also moving due to the rotation of the wheel in the opposite direction as the front point.
v_back = Velocity due to translation (10 m/s [E]) - Velocity due to rotation
v_back = 10 m/s [E] - (2π * 0.5 m * 10 m/s) [N]
Now, calculate the value:
v_back = 10 m/s [E] - 31.42 m/s [N]
You can calculate the magnitude and direction of this vector using the Pythagorean theorem and trigonometry. The magnitude is approximately 32.2 m/s, and the direction is approximately 17 degrees south of east.
So, in summary:
Top Point: 10 m/s [E]
Bottom Point: 10 m/s [E]
Front Point: Approximately 32.2 m/s, 17 degrees north of east
Back Point: Approximately 32.2 m/s, 17 degrees south of east