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Chapter 2: Problem 4

Consider two possible definitions of average speed: (a) the average of the values of the instantaneous speed over a time interval and (b) the magnitude of the average velocity. Are these definitions equivalent? Give two examples to demonstrate your conclusion.

Short Answer

Expert verified
No, these definitions are not equivalent. In cases where there is no change in direction, they may yield the same result. However, in cases with changes in direction, they can yield very different results.

Step by step solution

01

Understanding the Definitions

It's important to distinguish between speed and velocity to understand the question better. Speed is the rate of change of distance with respect to time, whereas velocity is the rate of change of displacement with respect to time. The magnitude (absolute value) of velocity could be different from speed, since velocity is a vector quantity which takes into account the direction of the motion, while speed doesn't.
02

Testing the Definitions

Consider a situation where a body travels in a straight line out and back. The average velocity would be zero because the displacement is zero, regardless of the distance traveled, hence the magnitude of the average velocity is zero. However, the average speed will not be zero, because some distance has been covered in that time.
03

Providing Examples

Example1: A person running on a circular track completes one round of 400m in 60s. Here, the average speed will be 400m divided by 60s which is around \(6.67m/s\) (definition a), whereas the magnitude of average velocity will be 0 (definition b). Example2: A car travels a straight 100m road in 20s then returns in another 20s. The average speed is \(5m/s\) while the magnitude of the average velocity is 0.

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