Problem 39 - Master Resource Book in JEE Main Physics

Question

A new unit of length is chosen such that the speed of light in vacuum is unity. Then the distance between the sun and the earth in terms of the new unit, if light takes 8 min and \(20 \mathrm{~s}\) to cover this distance? [NCERT] (a) 300 new unit of length (b) 500 new unit of length (c) 600 new unit of length (d) None of these

Step-by-Step Solution

Answer

The distance is 500 new units of length.

Step 1: Understand the Problem

We need to determine the distance between the sun and the earth given that the speed of light, which is usually approximately \( 3 \times 10^8 \text{ m/s} \), is equal to 1 in a new unit system. Light takes 8 minutes and 20 seconds to travel from the sun to the earth.

Step 2: Convert Time into Seconds

First, let's convert 8 minutes and 20 seconds into seconds. There are 60 seconds in one minute. Therefore, 8 minutes is \( 8 \times 60 = 480 \) seconds. Adding 20 more seconds gives us a total of \( 480 + 20 = 500 \) seconds.

Step 3: Determine the Distance with Unity Speed of Light

Since the speed of light is a unity measure (1) in the new unit, and it takes 500 seconds to reach the earth from the sun, the distance between the sun and the earth is simply the product of time and speed of light. Hence the distance in the new unit is \( 1 \times 500 = 500 \) new units of length.

Key Concepts

Unit Conversion

Unit conversion is a method in which we translate measurements from one set of units to another. This is particularly useful in physics where different problems might require different units. In our problem, we are dealing with a new, special unit where the speed of light is considered "unity," or simply the value of 1.

• The usual speed of light is approximately \(3 \times 10^8\text{ m/s}\).
• By setting the speed to be equal to 1 in the new unit, we simplify calculations significantly.

When the speed of light becomes 1, all distances measured over time are directly influenced by how many seconds light takes to travel that distance. Therefore, understanding the concept of unit conversion helps us easily calculate the new distance, as demonstrated in this exercise.

Time Measurement

Time measurement is crucial in physics calculations, especially when working with distances and speeds. Breaking down time into seconds is essential for simplifying and solving physics problems effectively.

• In our example, the time measurement starts as 8 minutes and 20 seconds.
• Converting this time into seconds makes calculations easier.

Steps in Time Conversion:

• 8 minutes translates to \(8 \times 60 = 480\) seconds.
• Add the remaining 20 seconds.
• Total time is \(480 + 20 = 500\) seconds.

Properly calculating time in seconds allows for seamless integration with other units and further calculations, underscoring its importance in physics.

Physics Problem Solving

Physics problem solving often involves breaking down problems into smaller, manageable parts. This is evident in problems dealing with universal constants, like the speed of light. Our task was to find the distance in a new unit where the speed of light is simplified to 1.

Problem-Solving Strategy:

• Understand the question: Determine what information is given and what is needed.
• Convert all quantities to compatible units, usually in their simplest form.
• Apply relevant formulas to find the solution, like the basic relation: distance = speed × time.

Since the new speed is unity (1), the distance is directly equal to the time taken by light to travel that distance. Here it's simply \(500\) new units because the new speed of light times the time taken \(1 \times 500\) results in \(500\). This showcases how simplifying units can clarify complex physics problems.