Question

Faster Trip. Suppose you wanted to reach Alpha Centauri in 100 years. (a) How fast would you have to go, in \(\mathrm{km} / \mathrm{hr}\) ? (b) How many times faster is the speed you found in (a) than the speeds of our fastest current spacecraft (around \(50,000 \mathrm{km} / \mathrm{hr}) ?\)

Short answer

To reach Alpha Centauri in 100 years, travel at approximately 47,150,000 km/hr, which is about 943 times faster than our fastest spacecraft.

Step-by-step solution

Understand the distance to Alpha Centauri

First, identify the distance from Earth to Alpha Centauri, which is approximately 4.367 light years. We need to convert this distance to kilometers. The distance of one light year is about 9.461 x 10^12 kilometers. Therefore, the distance to Alpha Centauri can be calculated as:\[4.367 \text{ light years} \times 9.461 \times 10^{12} \text{ km/light year} = 4.132 \times 10^{13} \text{ km}\]

Calculate speed in km/hr

We want to reach Alpha Centauri in 100 years. To find the speed required, divide the total distance by the total time.Convert 100 years into hours:\[100 \text{ years} \times 365.25 \text{ days/year} \times 24 \text{ hours/day} = 876600 \text{ hours}\]Now calculate the speed:\[\text{Speed} = \frac{4.132 \times 10^{13} \text{ km}}{876600 \text{ hours}} \approx 4.715 \times 10^7 \text{ km/hr}\]

Compare with the fastest current spacecraft

Now we need to find how much faster this speed is compared to the speed of the fastest spacecraft, which is 50,000 km/hr.Calculate the factor by dividing the speed from Step 2 by 50,000 km/hr:\[\text{Times faster} = \frac{4.715 \times 10^7 \text{ km/hr}}{50,000 \text{ km/hr}} \approx 943\]

Key concepts

Distance and Speed Calculations

When embarking on an interstellar journey, such as traveling to Alpha Centauri, understanding how to calculate distance and speed is crucial. The key here is to know the total distance and the time over which you want to travel.
This involves several basic concepts:

• Identifying the distance: In this case, Alpha Centauri is about 4.367 light years away from Earth.
• Converting time: You need to convert the desired travel time (in years) into hours to align with speed measurements typically expressed in kilometers per hour (km/hr).

For example, if you wish to reach Alpha Centauri in 100 years, you first need to figure out how many hours this period encompasses. Multiply 100 years by 365.25 days per year (accounting for leap years), and then by 24 hours per day. This results in a total of 876,600 hours.
The next step is to compute the necessary speed to cover that distance within the given time frame. Divide the total distance by the total time to find the speed. This process highlights how time and distance need to be measured in compatible units to accurately determine speed.

Light Year Conversion

In astrophysics, distances are vast, often requiring special units like light years for practicality. A light year is the distance light travels in a year, roughly equivalent to 9.461 x 10¹² kilometers. This unit simplifies the expression of astronomical distances.
To convert light years to kilometers for calculations, use this formula:

• Multiply the number of light years by 9.461 x 10¹² to find the distance in kilometers.

For example, for Alpha Centauri which is 4.367 light years away, the conversion is:\[4.367 \text{ light years} \times 9.461 \times 10^{12} \text{ km/light year} = 4.132 \times 10^{13} \text{ km}\]
Such conversions are essential in calculating how far we need to travel and understanding the enormous scale of space beyond Earth.

Spacecraft Speed Comparison

Once you've calculated the required speed for a cosmic journey, it’s insightful to compare it with current technological capabilities. Currently, the fastest spacecraft travel around 50,000 km/hr. This seems significant on Earth, but in the cosmos, it's quite modest.
To find out how much faster we would need to travel to reach Alpha Centauri in 100 years, calculate the factor by which this new speed exceeds our fastest spacecraft.

• Divide the required speed by the speed of existing spacecraft.
• For instance, if the calculated speed is approximately 4.715 x 10⁷ km/hr, the comparison is:\[\text{Times faster} = \frac{4.715 \times 10^7 \text{ km/hr}}{50,000 \text{ km/hr}} \approx 943\]

This means the speed must be 943 times faster than what's currently possible. Such comparisons illustrate the technological advances needed to explore interstellar space.