Question

Question

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?

Answer 1

Solution:
Hint

The instantaneous speed is the rate of change of the particle's position with respect to time. The instantaneous velocity is the vector representation of the instantaneous speed, which means that it includes both the magnitude and direction of the motion.

Step-by-step Solution:

1. Consider the definition of instantaneous speed and instantaneous velocity.

• Instantaneous speed: The rate of change of the particle's position with respect to time.

• Instantaneous velocity: The vector representation of the instantaneous speed, which means that it includes both the magnitude and direction of the motion.

2. Recognize that for infinitesimally small time intervals, the motion of the particle can be approximated to be uniform.

• This means that the particle will not have enough time to change its direction significantly during this time interval.

3. Therefore, the instantaneous velocity and instantaneous speed will have the same magnitude and direction.

Answer:

The instantaneous speed is always equal to the magnitude of instantaneous velocity because for infinitesimally small interval of time, the motion of the particle can be approximated to be uniform.

Note:

This is only an approximation, but it is very accurate for very small time intervals.