Question

The speed of light in vacuum is taken as unity. If light takes 6 min 40 s to reach the Earth from the Sun, the distance between the Sun and the Earth in new unit is: ____.

• A. 500
• B. 3 $\times$ 10⁸
• C. 400
• D. 3 $\times$ 10¹⁰

Hint:

This "new unit" is essentially a "light-second." One light-second is the distance light travels in one second. Since it takes 400 seconds, the distance is 400 light-seconds.

Answer 1

The Correct Option is C

Step 1: Understanding the Topic:
This problem pertains to the chapter "Units and Measurements," specifically focusing on the concept of derived units and the relationship between distance, speed, and time. In physics, we often use "Natural Units" where fundamental constants like the speed of light ($c$) are set to 1 to simplify calculations. This is common in relativistic physics and astronomy.
Step 2: Key Formulas and Approach:
The fundamental relationship used here is the kinematics equation for distance: \[ \text{Distance } (d) = \text{Speed } (v) \times \text{Time } (t) \] In this specific "new unit" system:
The speed of light ($c$) is defined as $1 \text{ unit}$.
Consequently, the unit of distance becomes the "light-second" (the distance light travels in one second).

Step 3: Detailed Explanation:

Analyze the given time: The time taken by light to travel from the Sun to the Earth is given as 6 minutes and 40 seconds. To perform calculations in a consistent system, we must convert this entire duration into seconds.
Time conversion:
6 minutes = $6 \times 60 = 360$ seconds.
Total time ($t$) = $360 \text{ s} + 40 \text{ s} = 400$ seconds.

Calculate the distance in new units: In the standard SI system, distance would be $c \times 400$. However, the problem states that the speed of light is unity ($c = 1$).
By substituting $v = 1$ and $t = 400$ into the distance formula: \[ \text{Distance} = 1 \times 400 = 400 \text{ units} \]
This implies that in a system where light travels at 1 unit per second, the distance is numerically equal to the time in seconds.
Step 4: Final Answer:
The distance between the Sun and the Earth in this new unit system is 400.