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Chapter 17: Problem 34

If the Crab Nebula has been expanding at an average velocity of \(1500 \mathrm{km} / \mathrm{s}\) since the year 1054 , what was its average radius in the year 2014 ? (Note: There are approximately \(3 \times 10^{7}\) seconds in a year.)

Short Answer

Expert verified
The average radius of the Crab Nebula in 2014 is \(4.32 \times 10^{13} \) km.

Step by step solution

01

- Calculate the Time Interval

Determine the number of years between 1054 and 2014. This is done by subtracting 1054 from 2014: \[ \text{Time Interval} = 2014 - 1054 = 960 \text{ years} \]
02

- Convert Years to Seconds

Convert the time interval from years to seconds using the given conversion factor.There are approximately \(3 \times 10^7\) seconds in a year. Multiply the number of years by the number of seconds per year: \[ \text{Time Interval in Seconds} = 960 \text{ years} \times 3 \times 10^7 \text{ seconds/year} = 2.88 \times 10^{10} \text{ seconds} \]
03

- Calculate the Average Radius

Use the average velocity and the total time to find the average radius.The radius is equal to the velocity multiplied by the time:\[ \text{Radius} = \text{Velocity} \times \text{Time Interval in Seconds} \]Substitute the given values:\[ \text{Radius} = 1500 \text{ km/s} \times 2.88 \times 10^{10} \text{ s} = 4.32 \times 10^{13} \text{ km} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

average velocity
To understand the expansion of the Crab Nebula, we first need to grasp the concept of average velocity. Average velocity is a measure of how quickly something moves from one point to another over a specific period. It is calculated by dividing the total distance traveled by the time it took to travel that distance. In this exercise, the Crab Nebula's average velocity is given as \(1500 \mathrm{km} / \mathrm{s}\). This means, on average, each second the nebula expands outward by 1500 kilometers.

Overall, average velocity helps us understand the rate of movement. Knowing the average velocity can also help us determine how far an object has traveled in a given time period.
time interval
To find out how much the Crab Nebula has expanded by 2014, we need to calculate the time interval since the nebula started expanding in 1054. A time interval is the amount of time between two events. In this case, the two events are the initiation of the expansion in 1054 and the measurement year of 2014.

We calculate the time interval by simple subtraction:
  • 2014 - 1054 = 960 years
So, the time interval over which we need to calculate the nebula’s expansion is 960 years.
unit conversion
For our calculation, we need to use consistent units. Given that the Crab Nebula's velocity is measured in kilometers per second, it helps to convert our time interval from years to seconds.

This step ensures that all units are consistent, making the subsequent calculations easier. Based on the problem, we use the conversion factor:
  • 1 year = \(3 \times 10^7\) seconds
We can now convert 960 years into seconds by multiplying:
  • 960 years \( \times 3 \times 10^7 \text{ seconds/year} = 2.88 \times 10^{10} \text{ seconds}\)
Now, our time interval is in a unit that aligns with our velocity unit.
astronomical calculations
With both the average velocity and the time interval in the correct units, we can now perform the final astronomical calculation to find the average radius of the Crab Nebula. This is done by multiplying the velocity by the time interval:

Equation: \( \text{Radius} = \text{Velocity} \times \text{Time Interval in Seconds} \)

Substituting the values:
  • \(\text{Radius} = 1500 \text{ km/s} \times 2.88 \times 10^{10} \text{ seconds} = 4.32 \times 10^{13} \text{ km}\)
This result means that by 2014, the average radius of the Crab Nebula has expanded to \(4.32 \times 10^{13}\) kilometers.

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