Problem 15 - Master Resource Book in JEE Main Physics

Question

One light year is defined as the distance travelled by light in one year. The speed of light \(3 \times 10^{8} \mathrm{~ms}^{-1}\). The same in metre is (a) \(3 \times 10^{12} \mathrm{~m}\) (b) \(9.461 \times 10^{15} \mathrm{~m}\) (c) \(3 \times 10^{15} \mathrm{~m}\) (d) None of these

Step-by-Step Solution

Answer

The correct option is (b) \(9.461 \times 10^{15} \text{ meters}\).

Step 1: Define a Light Year

A light year is the distance light travels in one year. This is used in astronomical calculations to measure vast distances.

Step 2: Determine the Speed of Light

The speed of light is given as \(3 \times 10^8 \text{ meters per second}\).

Step 3: Calculate Seconds in a Year

Calculate the total number of seconds in one year. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and approximately 365.25 days in a year (accounting for leap years). So, the total seconds in a year is:\[60 \times 60 \times 24 \times 365.25 = 31,557,600 \text{ seconds}\]

Step 4: Calculate the Distance in a Year

Multiply the speed of light by the number of seconds in a year to find the distance light travels in one year:\[3 \times 10^8 \text{ meters/second} \times 31,557,600 \text{ seconds} = 9.461 \times 10^{15} \text{ meters}\]

Step 5: Compare and Choose the Correct Option

The calculated distance is \(9.461 \times 10^{15} \text{ meters}\), which matches option (b).

Key Concepts

Speed of Light

Light travels at a mind-boggling speed, moving at approximately 299,792 kilometers per second, which is roughly equivalent to \(3 \times 10^8\) meters per second. To put this into perspective, if you were to travel around the Earth at this speed, you could circumnavigate the planet almost 7.5 times in just one second! This immense speed is foundational in the field of physics and essential for calculations involving astronomical distances.
Understanding the speed of light allows scientists to measure distances in space and time, making it a cornerstone concept in both astronomy and physics. It acts as a universal constant, aiding in the comprehension of the vast universe around us.

Astronomical Distance

When it comes to measuring distances in the universe, traditional units like kilometers or miles would become impractically large and cumbersome. Hence, astronomers use units like light years to measure these cosmic stretches.
A light year is the distance that light can travel in one year. This is equivalent to about 9.461 trillion kilometers (or \(9.461 \times 10^{15}\) meters).
Using light years provides a more understandable and simplified way to grasp the enormity of space distances. It helps scientists and astronomers express and compare the vastness of the universe more effectively, making it an indispensable tool in space science and exploration.

Unit Conversion

In science and everyday calculations, converting units is a crucial skill. It enables us to shift from one system of measurement to another seamlessly.
For astronomical calculations, converting the speed of light from meters per second to meters traveled per year involves using the total seconds in a year. Since the speed of light is a constant, this conversion helps in calculating large distances like a light year.
In this exercise, the speed of light is first expressed in meters per second and then multiplied by the number of seconds in a year to convert it to meters per year, resulting in a distance of \(9.461 \times 10^{15}\) meters. This demonstrates how powerful unit conversion can simplify complex scientific data into comprehensible metrics.

Seconds in a Year

To compute the number of seconds in a year, some straightforward multiplication is required. A minute consists of 60 seconds, an hour has 60 minutes, and a day has 24 hours. Multiplying these gives us the total seconds in a day. Then, considering there are approximately 365.25 days in a year (accounting for our leap year), we multiply again.
Therefore, the total number of seconds in a year calculates to:

• 60 seconds/minute
• 60 minutes/hour
• 24 hours/day
• 365.25 days/year

After doing the multiplication: \[60 \times 60 \times 24 \times 365.25 = 31,557,600\] seconds.
This number is instrumental when determining distances such as a light year, clearly illustrating how these basic components integrate to measure cosmic time frames.