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(a) How much time does it take light to travel from the moon to the earth, a distance of \(384,000 \mathrm{km}\) ? (b) Light from the star Sirius takes 8.61 years to reach the earth. What is the distance from earth to Sirius in kilometers?

Short Answer

Expert verified
(a) 1.28 seconds; (b) Approximately 8.14 x 10^13 km.

Step by step solution

01

Understand the Problem

We need to find two things: (a) the time it takes for light to travel from the Moon to Earth, given the distance, and (b) the distance from Earth to Sirius, given the time it takes light to travel that distance.
02

Use the Speed of Light for Part (a)

The speed of light is approximately \[ c = 299,792 \, \text{km/s} \]. We use the formula \[ \text{time} = \frac{\text{distance}}{\text{speed}} \] for part (a). Substitute the known values: \[ \text{time} = \frac{384,000 \, \text{km}}{299,792 \, \text{km/s}} \].
03

Calculate Time in Seconds

Perform the division to find the time it takes for light to travel from the Moon to Earth:\[ \text{time} \approx \frac{384,000}{299,792} \approx 1.28 \, \text{seconds} \].
04

Use the Known Light Travel Time for Part (b)

For part (b), we have the light travel time from Sirius to Earth as 8.61 years. The distance formula is \[ \text{distance} = \text{speed} \times \text{time} \]. First, we need to convert years to seconds.
05

Convert Years to Seconds

Given 8.61 years, we convert years to seconds:\[ 8.61 \, \text{years} \times 365.25 \, \text{days/year} \times 24 \, \text{hours/day} \times 60 \, \text{minutes/hour} \times 60 \, \text{seconds/minute} \approx 2.715 \times 10^8 \, \text{seconds} \].
06

Calculate the Distance to Sirius

Using the distance formula and the converted time:\[ \text{distance} = 299,792 \, \text{km/s} \times 2.715 \times 10^8 \, \text{seconds} \approx 8.14 \times 10^{13} \, \text{km} \].

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Key Concepts

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

Light Travel Time
Light travel time is a fascinating concept that helps us understand how fast light moves through space. Light travels at a constant speed of approximately 299,792 kilometers per second (km/s). To calculate how long light takes to travel from one point to another, we use the formula:
  • Time (seconds) = Distance (kilometers) / Speed of light (km/s)
For example, let's consider the distance from the Moon to Earth, which is about 384,000 km. Using our formula, we substitute the values:
  • Time ≈ 384,000 km / 299,792 km/s ≈ 1.28 seconds
It means that it takes about 1.28 seconds for light to travel from the Moon to the Earth. This quick journey helps us comprehend the vast speed at which light travels through space and the concept of *light travel time* gives us insight into the distances of celestial bodies from Earth.
Distance Calculations
Distance calculations are crucial in astronomy, allowing us to measure how far celestial objects are from Earth. One of the key steps in solving these calculations involves understanding how to use the speed of light to determine distance. We often use the formula:
  • Distance = Speed of light × Time
This formula helps when you know the time it takes light to travel from a star to our planet. For instance, light from the star Sirius takes 8.61 years to reach Earth. To find the distance in kilometers, we first convert 8.61 years into seconds. Knowing there are approximately 31,557,600 seconds in a year, we calculate:
  • Seconds = 8.61 years × 31,557,600 seconds/year ≈ 271,465,696 seconds
Then we use our distance formula:
  • Distance = 299,792 km/s × 271,465,696 s ≈ 8.14 × 10¹³ kilometers
Through these calculations, we unveil the vast distances across space with ease, converting light-years into kilometers to understand the scale of the universe.
Astronomical Distances
Astronomical distances are incredibly vast and often difficult to fathom compared to distances we encounter on Earth. These cosmic distances are usually measured in terms of light-years, a unit representing the distance light travels in one year. To grasp these distances in more tangible units, such as kilometers, we need key conversions. In astronomy:
  • 1 light-year = 9.461 trillion kilometers (approximately)
For our calculations with Sirius, a well-known star, the light-year distance simplifies to astronomical units useful for understanding the scale of space. With our previous calculation, knowing that light takes 8.61 years to reach us, it helps put the universe in perspective without needing a calculator to visualize just how far Learning about astronomical distances aids us in mapping out our universe. It shows how interconnected time and space are, as seen in the light travel time and quantified in distance calculations. Comprehending these vast cosmic distances allows us to appreciate the universe's immensity and the intricacy of celestial navigation.

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Most popular questions from this chapter

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