In Exercises 2.9 and 2.10, we have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?
Instantaneous speed is always equal to the magnitude of instantaneous velocity because when we consider an infinitesimally small time interval, the distinction between path length and displacement vanishes.
For average quantities over a finite time interval, an object may change its direction of motion. This means the total path length can be greater than the magnitude of displacement. Hence, average speed (total path length divided by time) can be greater than or equal to the magnitude of average velocity (displacement divided by time).
However, for instantaneous quantities, the time interval Δt approaches zero. In such an infinitesimally small interval, the object cannot reverse its direction. It moves in essentially one direction only, making the path length equal to the magnitude of displacement. Therefore, instantaneous speed equals the magnitude of instantaneous velocity at that instant.
Explanation
The textbook explicitly states in Section 2.2 that 'instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.' The key mathematical concept is that velocity is defined as the limit of average velocity as Δt → 0. In this infinitesimal limit, the particle cannot change direction, so distance traveled equals magnitude of displacement. This differs from finite intervals where direction changes are possible, causing average speed to potentially exceed magnitude of average velocity.