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 $=$ $3.00\times {10}^{\mathrm{s}}\mathrm{m}/\mathrm{s};1\mathrm{knot}=2000\mathrm{yd}/\mathrm{h}$, exactly. $)$

Short answer

The U.S.S. Enterprise, traveling at warp factor 1.71, has a speed of approximately $2.92\times {10}^9$ knots and $3.36\times {10}^9$ miles per hour.

Step-by-step solution

Calculate the speed of the Enterprise in meters per second

First, we need to calculate the speed of the U.S.S. Enterprise in meters per second, knowing that warp 1.71 corresponds to 5 times the speed of light. The speed of light is given as $3.00\times {10}^8$ meters per second. Therefore, the speed of the Enterprise at warp factor 1.71 is: Enterprise speed = Warp factor * Speed of light = $5.00\ast (3.00\times {10}^8)m/s$ Enterprise speed = $1.50\times {10}^{9}m/s$

Convert the speed from meters per second to knots

We will first convert the speed of the Enterprise from meters per second to knots. The conversion is provided as $1knot=2000yd/h$. First, we need to convert 1 knot to meters per second: 1 mile = 1760 yards, so 1 yard = $1/1760$ mile 1 knot = $2000\ast \frac{1}{1760}mph$ Now, we know that 1 mile = 1,609.34 meters and 1 hour = 3,600 seconds. So, we can convert the above to meters per second: 1 knot = $2000\ast \frac{1}{1760}\ast 1609.34m/h$ 1 knot = $2000\ast \frac{1}{1760}\ast 1609.34\ast \frac{1}{3600}m/s$ 1 knot ≈ 0.514444\,m/s Now, we can convert the speed of the Enterprise to knots: Enterprise speed in knots = $\frac{1.50\times {10}^9}{0.514444}knots$ Enterprise speed in knots ≈ $2.92\times {10}^{9}knots$

Convert the speed from meters per second to miles per hour

Next, we will convert the speed of the Enterprise from meters per second to miles per hour using the conversions we have already calculated: 1 mph =$\frac{1609.34m/h}{3600s/h}$ 1 mph ≈ 0.44704\,m/s Now, we can convert the speed of the Enterprise to miles per hour: Enterprise speed in mph = $\frac{1.50\times {10}^9}{0.44704}mph$ Enterprise speed in mph ≈ $3.36\times {10}^{9}mph$

Final Answer:

The U.S.S. Enterprise, traveling at warp factor 1.71, has a speed of approximately $2.92\times {10}^9$ knots and $3.36\times {10}^9$ miles per hour.

Key concepts

Speed of Light

The concept of the speed of light is crucial in the realm of physics and a common measure in discussing astronomical distances or theoretical concepts, such as warp speed in science fiction. The speed of light in a vacuum is approximately $3.00\times {10}^{8}m/s$. This constant is not just a speed limit for light but for all matter and information in the universe. In exercises where velocity needs to be compared with the speed of light, it is essential to understand that reaching or exceeding this speed with current technology remains within the bounds of science fiction.

Additionally, the speed of light is an underpinning factor in Einstein's theory of relativity, specifically affecting time dilation and the contraction of lengths when approaching light speed. Understanding the speed of light not only helps solve problems in a physics context but also piques curiosity about the fundamental laws that govern our universe.

Metric to Imperial Conversion

Converting between metric and imperial units is a common requirement in scientific and everyday calculations. The metric system, which includes units like meters and seconds, is universally used in scientific measurements due to its simplicity and widespread international recognition. On the other hand, the imperial system uses units such as miles, yards, and hours, and is still prevalent in some countries for everyday applications.

To navigate between the two systems, you must know the equivalent values, such as 1 mile equaling approximately 1,609.34 meters or 1 yard being equivalent to 0.9144 meters. These conversions are the backbone of solving problems that involve units from different systems, ensuring accurate and contextual understandings of measurements, such as when converting speed from meters per second (metric) to miles per hour (imperial).

Key Conversion Factors

• 1 mile = 1,609.34 meters
• 1 yard = 0.9144 meters
• 1 hour = 3,600 seconds

The ability to efficiently convert between these units is an essential skill for students and professionals dealing with measurements, travel, physics, and numerous other fields.

Mathematical Problem Solving

Mathematical problem solving involves a systematic process to understand and find solutions to complex problems. It starts with comprehension of the problem, identifying what is known and what needs to be determined. The next step is developing a strategy, like establishing intermediate goals or breaking down the problem into more manageable parts.

In the realm of physics or real-world scenarios, these strategies also include unit conversion, algebraic manipulation, and applying relevant physical laws or constants. Mathematical problem solving encourages logical reasoning and critical thinking, skills that are invaluable not just within mathematics, but in everyday life and the pursuit of scientific knowledge. To improve competency in solving mathematical problems, practice is key, along with reviewing concepts and methodologies for a more in-depth understanding.

Effective problem solvers also check their solutions, reviewing each step to ensure no detail is overlooked. This step is significant for accuracy and in developing confidence in one’s ability to tackle similar or more challenging problems in the future.