This problem involves calculating the time it would take for the New Horizons probe to travel to Proxima Centauri, located about 4.25 light-years away, at a constant speed of 14 km/s. Here's how we can approach it:

Step 1: Convert the distance into kilometers

The distance to Proxima Centauri is 4.25 light-years. To proceed, we need to convert this distance into kilometers. The conversion factor is: 1 light-year = 9.461 × 10¹² km

Therefore: $4.25 \, \text{light-years} = 4.25 \times 9.461 \times 10^{12} \, \text{km}$ $= 4.015 \times 10^{13} \, \text{km}$

Step 2: Use the speed to calculate the time

Now that we have the distance in kilometers and the speed in kilometers per second (14 km/s), we can use the formula for time:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Substituting the values:

$\text{Time} = \frac{4.015 \times 10^{13} \, \text{km}}{14 \, \text{km/s}}$ $\text{Time} \approx 2.87 \times 10^{12} \, \text{seconds}$

Step 3: Convert time into years

Since the time is in seconds, we need to convert it into years. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365.25 days in a year (taking into account leap years).

$1 \, \text{year} = 60 \times 60 \times 24 \times 365.25 \, \text{seconds} = 31,557,600 \, \text{seconds}$

Now, we can convert the time from seconds to years:

$\text{Time in years} = \frac{2.87 \times 10^{12} \, \text{seconds}}{31,557,600 \, \text{seconds/year}}$ $\text{Time in years} \approx 91,000 \, \text{years}$

Final Answer:

It would take approximately 91,000 years for the New Horizons probe to reach Proxima Centauri if it maintained a constant speed of 14 km/s.

Let me know if you'd like further clarification or if you'd like to work through additional questions!

Here are some related questions you might find interesting:

1. How would the time change if the speed of the probe were doubled?
2. How can we compare the travel time to other spacecraft missions (e.g., Voyager)?
3. What other methods could be used to increase spacecraft speeds over long distances?
4. How would the time to reach Proxima Centauri change if we were considering the speed of light?
5. What factors might affect the actual travel time of the New Horizons probe?

Tip: When working with very large distances like light-years, always convert all units to be consistent (e.g., kilometers) to simplify calculations.