Question

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

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

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

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 = \(5.145 \times 10^9 \mathrm{m/s}\)

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 = \(\frac{5.145 \times 10^9}{2030 \times 0.9144 \times \frac{1}{3600}} \mathrm{knots}\)

4. Calculate the speed in knots

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

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 = \(\frac{5.145 \times 10^9}{\frac{1}{1609.34} \times 3600} \mathrm{mph}\)

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.

Key concepts

Speed of Light

The speed of light is one of the most fundamental constants in physics. It travels at an astounding rate of approximately ! \(3.00 \times 10^8 \) meters per second (m/s). This means light can circle the entire Earth about 7.5 times in just one second! It's hard to fully comprehend just how fast this is, but it's crucial for many calculations in physics and astronomy. In science fiction, the speed of light often serves as a benchmark for areasonable speed limit in the universe. However, fictional narratives like Star Trek often exceed it, inventing concepts like 'warp speed.' When the U.S.S. Enterprise travels at warp 1.71, the ship is moving at 5 times the speed of light, illustrating how this speed becomes an exciting tool in narrative settings.

Unit Conversion

Unit conversion is vital in making sense of measurements across different systems, whether for simple recipes or complex physics problems. In this scenario, we needed to convert from meters per second, the speed of the starship, to other units like knots and miles per hour. Let's break it down: - **Knots**: This unit originally comes from nautical speed measurements, where 1 knot is equal to roughly 2030 yards per hour. To convert meters per second to knots, we transform meters to yards (as 1 meter is approximately 0.9144 yards) and seconds to hours (3600 seconds in an hour). - **Miles per Hour**: Familiar in everyday conversations, converting from meters per second to miles per hour involves knowing that 1 mile is approximately 1609.34 meters. Then, adjusting for hours using the same principle as above, we arrive at the needed speed in mph. Altogether, these conversions help in comparing speeds across diverse contexts, highlighting the profound role that unit conversions play in scientific calculations and real-world applications.

Science Fiction Spaceship Speed

In the realm of science fiction, speed is not just a number, it's an instance of imagination meeting physics. Sci-fi spaceships often travel faster than light, creating a thrilling exploration theme in stories. In Star Trek, for instance, 'warp speed' lets ships navigate vast interstellar distances quickly, defying the constraints posed by the real speed of light. At warp factor 1.71, the U.S.S. Enterprise, for example, moves at 5 times the speed of light. This fictional speed allows for imaginative storytelling, giving viewers a sense of the infinite possibilities in space travel. Science fiction uses these advanced speeds to challenge our understanding of physics, encouraging questions about space, time, and technological possibilities. At its heart, it's less about the exact numbers and more about envisioning a limitless universe for exploration.