Question

The star Arcturus is \(3.50 \times 10^{14} \mathrm{~km}\) from the earth. How many days does it take for light to travel from Arcturus to earth? What is the distance to Arcturus in light years? One light year is the distance light travels in one year \(\left(365\right.\) days); light travels at a speed of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\).

Short answer

Light from Arcturus takes approximately 1,233.6 days to reach Earth, which is about 3.38 light years away.

Step-by-step solution

Convert distance from kilometers to meters

Since speed of light is given in meters per second, and the distance is given in kilometers, convert the distance from kilometers to meters by multiplying by 1,000 (since 1 km = 1,000 m).

Calculate the time in seconds

Use the formula, Time = Distance / Speed, to calculate the time it takes for light to travel from Arcturus to Earth. Remember to use the distance in meters and the speed of light in meters per second.

Convert time from seconds to days

There are 86,400 seconds in a day (60 seconds per minute, 60 minutes per hour, 24 hours per day). Convert the time from seconds to days by dividing by 86,400.

Calculate the distance in light years

Since 1 light year is the distance light travels in one year, to find the distance in light years, divide the distance the light travels from Arcturus to Earth by the distance light travels in one year.

Convert time from days to years for light year calculation

To find out the distance in light years, we need to know how many years it takes for light to travel the given distance. Since we know the time in days from Step 3, we will convert that into years by dividing by 365 (for simplicity, ignoring leap years).

Key concepts

Light Year Conversion

Understanding light year conversion plays a crucial role in measuring immense distances in astronomy. A light year is a measure of distance, not time, defined as the length that light, which moves at an unfathomably fast pace, travels within one Earth year.

To convert a distance from kilometers to light years, a two-step process is often followed. First, the distance in kilometers is converted to meters since our given speed of light is in meters per second (\(3.00 \times 10^{8}\text{ m/s}\)). This step ensures homogeneous units for the calculation. Once the distance is in meters, you determine how many of those meters equal the total distance that light travels in a year. Since light moves at \(3.00 \times 10^{8}\text{ m/s}\), multiplying this speed by the number of seconds in a year gives you one light year in meters. The distance from Arcturus to Earth in kilometers can thus be expressed as a fraction of this vast measure, seamlessly putting vast cosmic distances into perspective.

For instance, in this case, once the distance from Arcturus to Earth is expressed in meters, it is divided by the distance light travels in a year. The result offers an enhanced understanding of the staggering scale of space, as we express these distances in terms relative to the speed of light.

Scientific Notation

Scientific notation is a way of writing very large or very small numbers concisely.

It's a format that expresses numbers as a product of two parts: a coefficient and 10 raised to a power. The coefficient is a number greater than or equal to 1 and less than 10, and the power of 10 indicates how many places to move the decimal point. In our exercise, the distance to Arcturus is given as \(3.50 \times 10^{14} \text{ km}\), which is much more manageable than writing out 350,000,000,000,000 kilometers!

When dealing with astronomical distances or the incredibly fast speed of light, scientific notation is indispensable. It simplifies calculations and helps avoid mistakes that could arise from handling many zeros in a standard decimal format.

Speed of Light

The speed of light is a fundamental constant in the universe, crucial for understanding how distance and time relate in space. It's generally denoted by the symbol \(c\), and in a vacuum, it measures approximately \(3.00 \times 10^{8} \text{ m/s}\). When we say 'speed of light', we're referring to its velocity in a vacuum, which is different from how light travels through mediums like water or glass where it is slower.

Why Is It Important?

The speed of light is central to calculations in astronomy and physics. It's not only a speed limit for matter, but also the medium through which we learn about the universe. For example, when we receive light from distant stars, it's akin to looking back in time; that light has traveled at \(c\) for many years to reach us.

In our exercise, we reliance on the constant speed of light allows us to calculate how long it would take light to travel from Arcturus to our planet.

Time-Distance-Speed Relationship

The time-distance-speed relationship is a fundamental principle in physics, applicable in various contexts, from everyday motion to the cosmic scale of space travel. It's encapsulated in the simple formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

This formula allows us to solve for any one of the three variables, provided the other two are known. In the context of light traveling from a star to Earth, we calculate time by knowing the distance light needs to cover and the speed at which it travels.

Applying the Principle

When we apply this relationship to our exercise involving Arcturus, we start by converting the given distance to meters, to match the unit used for the speed of light. Next, we divide this distance by the speed of light to get the time in seconds. Finally, to make sense of this time for a human perspective, we convert seconds to days and potentially to years, giving us a more comprehensible measure of the vast distances light covers across the cosmos.