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Chapter 1: Problem 64

Shannon decides to check the accuracy of her speedometer. She adjusts her speed to read exactly 70 mph on her speedometer and holds this steady, measuring the time between successive mile markers separated by exactly 1.00 mile. If she measures a time of \(54 \mathrm{s},\) is her speedometer accurate? If not, is the speed it shows too high or too low?

Short Answer

Expert verified
The speed displayed on the speedometer is too high, indicating the speedometer is not accurate.

Step by step solution

01

Convert the Speed to SI units

First, convert the speed of 70 mph to SI units (meters per second), as the given time is in seconds and the distance in miles. The conversion factors are: 1 mile = 1609.34 meters, and 1 hour = 3600 seconds. Therefore, the speed in m/s is \((70*1609.34)/3600\).
02

Calculate the real speed

Next, calculate the real speed using the formula \(Speed = Distance/Time\). The distance between mile markers is exactly 1.00 mile which is 1609.34 meters, and the time measured is 54 seconds. Hence, the real speed is \(1609.34/54\).
03

Compare the two speeds

Now, compare the speed from the speedometer (converted to m/s in step 1) with the real speed calculated in step 2. If they are the same, the speedometer is accurate. If not, determine which one is higher to specify whether the speedometer overestimates or underestimates the speed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Unit conversion is a fundamental skill in physics and engineering. It allows you to switch from one measurement system to another. In Shannon's case, she needed to convert her speed from miles per hour (mph) to meters per second (m/s).
To perform this conversion, you can use specific factors:
  • 1 mile equals 1609.34 meters
  • 1 hour equals 3600 seconds
So, to convert 70 mph into m/s:
Multiply 70 by 1609.34 to change miles into meters. Then divide by 3600 to turn hours into seconds.
This gives the formula: \[\text{Speed in m/s} = \frac{70 \times 1609.34}{3600}\] Using unit conversion correctly ensures that you can calculate speeds and other measurements accurately across different systems.
Speed Calculation
Calculating speed is about understanding the relationship between distance and time. Shannon's speed calculation required knowing how fast she traveled over a known distance in a specified amount of time.
The formula for speed is:
\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]In Shannon's situation, she traveled 1.00 mile between markers, which converts to 1609.34 meters. With a measured time of 54 seconds, the real speed calculation looked like this:
\[\text{Real Speed} = \frac{1609.34}{54}\]Making such calculations helps us figure out whether the speedometer reading is accurate or if adjustments are needed.
Comparison of Speeds
Once you have both the speedometer's reading (converted to m/s) and the real speed, the next step is to compare them.
This comparison is straightforward:
  • If the speeds match, the speedometer is accurate.
  • If they don't match, you'll find that one speed is higher or lower than the other.
In Shannon's case, if the speed from the speedometer (from our unit conversion step) does not equal the real speed (from our calculation), then it's time to ask whether the discrepancy is an overestimation or underestimation.
Determining the accuracy of the speedometer involves checking which speed is greater. If the speedometer's speed is higher, it overestimates; if it's lower, it underestimates the true speed. This aspect of the exercise cultivates analytical thinking and enhances attention to detail.

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Most popular questions from this chapter

John walks \(1.00 \mathrm{km}\) north, then turns right and walks \(1.00 \mathrm{km}\) east. His speed is \(1.50 \mathrm{m} / \mathrm{s}\) during the entire stroll. a. What is the magnitude of his displacement, from beginning to end? b. If Jane starts at the same time and place as John, but walks in a straight line to the endpoint of John's stroll, at what speed should she walk to arrive at the endpoint just when John does?

If you swim with the current in a river, your speed is increased by the speed of the water; if you swim against the current, your speed is decreased by the water's speed. The current in a river flows at \(0.52 \mathrm{m} / \mathrm{s}\). In still water you can swim at \(1.78 \mathrm{m} / \mathrm{s} .\) If you swim downstream a certain distance, then back again upstream, how much longer, in percent, does it take compared to the same trip in still water?

Estimate the length of a human lifetime, in seconds.

A car travels along a straight east-west road. A coordinate system is established on the road, with \(x\) increasing to the east. The car ends up 14 mi west of the origin, which is defined as the intersection with Mulberry Road. If the car's displacement was -23 mi, what side of Mulberry Road did the car start on? How far from the intersection was the car at the start?

I If you make multiple measurements of your height, you are likely to find that the results vary by nearly half an inch in either direction due to measurement error and actual variations in height. You are slightly shorter in the evening, after gravity has compressed and reshaped your spine over the course of a day. One measurement of a man's height is 6 feet and 1 inch. Express his height in meters, using the appropriate number of significant figures.
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