Chapter 1: Problem 21

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: The GPS receiver experiences linear acceleration in scenarios a and b, and experiences centripetal acceleration in scenario c.

Step by step solution

Initial and Final Velocity Vectors a:

In this case, the initial velocity (v1) is \(60 mph\) at \(70^{\circ}\) from east. Using a compass, this means the driver is going northeast. The final velocity (v2) is \(50 mph\), and the direction remains the same, \(70^{\circ}\).

Determine the Change in Velocity a:

To determine the change in velocity (Δv), we will look at the magnitude and direction. The magnitude of the velocity has decreased from \(60 mph\) to \(50 mph\), which means magnitude of acceleration is the difference between the two velocities, \(10 mph\). The direction remains the same, so there is only a change in speed and not direction.

Acceleration a:

With only a change in speed and not direction, the receiver is experiencing linear acceleration, slowing down in the same direction of travel. Scenario b:

Initial and Final Velocity Vectors b:

In this case, the initial velocity (v1) is \(50 mph\) at \(70^{\circ}\) from east, again going northeast. The final velocity (v2) is \(70 mph\), and the direction remains the same, \(70^{\circ}\).

Determine the Change in Velocity b:

The magnitude of the velocity has increased from \(50 mph\) to \(70 mph\), leading to an acceleration of \(20 mph\). The direction is the same.

Acceleration b:

As only the speed has changed and not the direction, the receiver is experiencing linear acceleration, speeding up in the same direction of travel. Scenario c:

Initial and Final Velocity Vectors c:

In this case, the initial velocity (v1) is \(60 mph\) at \(70^{\circ}\) from east, going northeast. The final velocity (v2) is also \(60 mph\), but the direction has changed to \(90^{\circ}\)

Determine the Change in Velocity c:

The speed stays the same, but the direction has changed from \(70^{\circ}\) to \(90^{\circ}\), which means the receiver is now moving due east.

Acceleration c:

Since the speed remains constant, but the heading has changed, the receiver is experiencing centripetal acceleration, moving around a curve with a constant speed. In summary, the receiver is accelerating in all three scenarios. Scenario a experiences linear acceleration while slowing down in the same direction. Scenario b experiences linear acceleration while speeding up in the same direction. Scenario c experiences centripetal acceleration as it maintains a constant speed while changing direction.

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Short Answer

Step by step solution

Initial and Final Velocity Vectors a:

Determine the Change in Velocity a:

Acceleration a:

Initial and Final Velocity Vectors b:

Determine the Change in Velocity b:

Acceleration b:

Initial and Final Velocity Vectors c:

Determine the Change in Velocity c:

Acceleration c:

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