Final answer:
The magnitudes of instantaneous velocity and instantaneous speed are equal because they represent the rate of motion at a specific instant without regard to direction. Average speed differs from the magnitude of average velocity because it involves total distance traveled versus displacement, especially evident in curved paths.
Step-by-step explanation:
The magnitudes of instantaneous velocity and instantaneous speed are always equal because they describe the motion of an object at a particular moment in time. Instantaneous speed is the magnitude of instantaneous velocity, meaning it is the absolute value of the velocity vector and does not include direction, just the rate of motion.
However, average speed and the magnitude of average velocity are different when considering motion over a period of time along a curved path. Average speed is calculated by dividing the total distance traveled by the elapsed time, whereas magnitude of average velocity is calculated by dividing the total displacement (straight-line distance from the starting point to the final position) by the elapsed time. Because displacement can be less than the total path length traveled (especially for curved paths), the average speed can be greater than the magnitude of average velocity.
For example, if an object moves in a circular path and returns to its starting point, the displacement is zero, and hence the average velocity is zero, but the average speed is not zero because the object has covered a distance around the circle.