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Chapter 1: Problem 23

The following are speeds and headings displayed on a GPS receiver. (Heading gives the direction of motion based on north \(=0^{\circ}\), east = \(90^{\circ}\), south \(=180^{\circ}\), etc.) In each case, indicate whether the receiver was accelerating during the time between the displays and, if it was, describe in what way the receiver was accelerating. (a) Initially: \(60 \mathrm{mph}, 70^{\circ} .5\) seconds later: \(50 \mathrm{mph}, 70^{\circ} .\) (b) Initially: \(50 \mathrm{mph}, 70^{\circ} .5\) seconds later: \(70 \mathrm{mph}, 70^{\circ}\). (c) Initially: \(60 \mathrm{mph}, 70^{\circ} .5\) seconds later: \(60 \mathrm{mph}, 90^{\circ}\).

Short Answer

Expert verified
Answer: A GPS receiver experiences acceleration in all three cases: (a) when the speed decreases and direction remains the same, (b) when the speed increases and direction remains the same, and (c) when the speed remains the same but the direction changes. Acceleration can occur due to changes in speed, direction, or both.

Step by step solution

01

Case (a) Solution:

In this scenario, the GPS receiver's speed decreased from 60 mph to 50 mph, while the direction remained constant at 70 degrees. Because there was a change in speed, even though the direction remained the same, there was still acceleration. This instance demonstrates acceleration in speed (negative or decreasing acceleration, specifically).
02

Case (b) Solution:

Initially, the GPS receiver's speed increased from 50 to 70 mph while the direction remained constant at 70 degrees. Similar to the first case, there was a change in speed but not in direction. Hence, there was acceleration. This instance demonstrates acceleration in terms of increasing speed (positive acceleration).
03

Case (c) Solution:

In this case, the receiver's speed remained the same at 60 mph, but the direction changed from 70 to 90 degrees. Hence, there was a change in the direction of motion but not in speed. This instance demonstrates acceleration in direction (angular or centripetal acceleration). Therefore, a GPS receiver can accelerate by changing either its speed or its direction of motion. A change in either or both of these elements indicates that acceleration has taken place.

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