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Chapter 1: Problem 53

Science fiction often uses nautical analogies to describe space travel. If the starship U.S.S. Enterprise is traveling at warp factor \(1.71,\) what is its speed in knots and in miles per hour? (Warp \(1.71=5.00\) times the speed of light; speed of light = \(3.00 \times 10^{8} \mathrm{m} / \mathrm{s} ; 1\) knot \(=2030 \mathrm{yd} / \mathrm{h} .\) )

Short Answer

Expert verified
The starship U.S.S. Enterprise is traveling at approximately \(8,682,586.88 \mathrm{knots}\) or \(11,530,000 \mathrm{mph}\) at warp factor 1.71.

Step by step solution

01

1. Find U.S.S. Enterprise's speed in meters per second

To find the starship's speed at warp factor 1.71, we use the given relationship: Warp 1.71 = 5.00 times the speed of light We also know the speed of light is \(3.00 \times 10^8 \mathrm{m/s}\). Now, we can calculate the speed of the U.S.S. Enterprise in meters per second: Speed = (Warp factor) × (Speed of light) = 5.00 × \(3.00 \times 10^8\) × 1.71 meters per second
02

2. Calculate the speed in meters per second

Now plug the values into the expression and calculate the speed in meters per second: Speed = 5.00 × \(3.00 \times 10^8\) × 1.71 = \(5.145 \times 10^9 \mathrm{m/s}\)
03

3. Convert meters per second to knots

Now, let's convert the speed from meters per second to knots. We know 1 knot = \(2030 \mathrm{yd/h}\) and 1 yard = 0.9144 meters. We first convert yards per hour to meters per hour: 1 knot = 2030 yd/h × 0.9144 m/yd We then convert meters per hour to meters per second: 1 knot = (2030 × 0.9144) m/h × 1h/3600s Now we divide the speed of the U.S.S. Enterprise in meters per second by the conversion factor for knots: Speed in knots = \(\frac{5.145 \times 10^9}{2030 \times 0.9144 \times \frac{1}{3600}} \mathrm{knots}\)
04

4. Calculate the speed in knots

Now plug the values into the expression and calculate the speed in knots: Speed in knots = \(\frac{5.145 \times 10^9}{2030 \times 0.9144 \times \frac{1}{3600}} = 8682586.88 \mathrm{knots}\)
05

5. Convert meters per second to miles per hour

Next, we'll convert the speed from meters per second to miles per hour. We know 1 mile = 1609.34 meters. First, convert meters per second to miles per second: 1 mile/s = \(\frac{1}{1609.34}\) m/s Now, convert miles per second to miles per hour: 1 mile/h = 3600 mile/s Now we divide the speed of the U.S.S. Enterprise in meters per second by the conversion factor for miles per hour: Speed in miles per hour = \(\frac{5.145 \times 10^9}{\frac{1}{1609.34} \times 3600} \mathrm{mph}\)
06

6. Calculate the speed in miles per hour

Now plug the values into the expression and calculate the speed in miles per hour: Speed in miles per hour = \(\frac{5.145 \times 10^9}{\frac{1}{1609.34} \times 3600} = 1.153 \times 10^7 \mathrm{mph}\) To conclude, the starship U.S.S. Enterprise is traveling at approximately \(8,682,586.88 \mathrm{knots}\) or \(11,530,000 \mathrm{mph}\) at warp factor 1.71.

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