Q.The distance of a galaxy is of the order of $10^{25}$ m. Calculate the order of magnitude of time taken by light to reach us from the galaxy.

The core concept here is the relationship between speed, distance, and time. Light travels at a constant speed in a vacuum, allowing us to calculate the time taken to cover a given distance.

1. The given distance of the galaxy is $D \approx 10^{25}$ m.
2. The speed of light in a vacuum is $c \approx 3 \times 10^8$ m/s. For order of magnitude calculations, we can consider its order as $10^8$ m/s.
3. Substitute these values into the formula: $T = \frac{10^{25} \text{ m}}{3 \times 10^8 \text{ m/s}} = \frac{1}{3} \times 10^{25-8} \text{ s} = \frac{1}{3} \times 10^{17} \text{ s}$. …

When we talk about the "order of magnitude" of a quantity, we are essentially asking for the power of 10 that best represents its scale. It's a way to express very large or very small numbers in a simplified form, focusing on their scale rather than their precise value. For instance, $5 \times 10^3$ and $8 \times 10^3$ both have an order of magnitude of $10^3$, even though their exact values differ.

In this problem, we are given the distance to a galaxy and asked for the order of magnitude of time light takes to travel that distance. This is a classic application of the fundamental relationship between speed, distance, and time. Light travels at a constant speed in a vacuum, so if we know the distance and the speed, we can directly calculate the time.

Let's break down the calculation step-by-step:

1. Identify the given distance:

   The distance of the galaxy is given as $10^{25}$ m. This is already expressed in a convenient power-of-10 form, which is ideal for order of magnitude calculations.

   $D = 10^{25} \text{ m}$

2. Recall the speed of light:

   Light travels at a constant speed in a vacuum, denoted by $c$. Its approximate value is $3 \times 10^8$ meters per second.

   $c = 3 \times 10^8 \text{ m/s}$

3. Calculate the time taken:

   Using the formula $T = D/c$, we substitute the values:

   $T = \frac{10^{25} \text{ m}}{3 \times 10^8 \text{ m/s}}$

   $T = \frac{1}{3} \times \frac{10^{25}}{10^8} \text{ s}$

   When dividing powers of 10, we subtract the exponents:

   $T = \frac{1}{3} \times 10^{(25-8)} \text{ s}$

   $T = \frac{1}{3} \times 10^{17} \text{ s}$

4. Convert the fraction to a decimal and express in scientific notation:

   $\frac{1}{3}$ is approximately $0.333...$. So,

   $T \approx 0.333 \times 10^{17} \text{ s}$

   To express this in standard scientific notation (where the coefficient is between 1 and 10), we move the decimal point one place to the right and adjust the exponent: …

Method: Dimensional Analysis Using Speed-Distance-Time Relation

Concept: The fundamental relation connecting speed, distance, and time is:

Rearranging for time:

Steps

Step 1: Identify the given data

• Distance of galaxy, $d = 10^{25}$ m (order of magnitude)
• Speed of light, $c = 3 \times 10^8$ m/s (standard value)

Step 2: Write the formula for time

Step 3: Substitute the values (using orders of magnitude)

Step 4: Simplify the powers of 10

Step 5: Find the order of magnitude

• $\frac{1}{3} \approx 0.33$, which is less than $3.16$ (i.e., $\sqrt{10}$) …

Here are the most common mistakes students make on this Speed-Distance-Time problem (especially in the context of orders of magnitude and scientific notation), along with how to avoid each.

Mistake 1: Using the wrong formula or rearranging incorrectly

The error:

Students often confuse the relationship between speed, distance, and time. They might multiply when they should divide, or use $v = d \times t$ instead of $t = \frac{d}{v}$.

How to avoid:

Always write the triangle or the basic formula first:

Rearrange for time:

For light, speed $c = 3 \times 10^8 \ \text{m/s}$. So:

Key check: If the answer comes out smaller than 1 second for a galaxy, you’ve inverted the formula.

Mistake 2: Forgetting the speed of light value or using an incorrect one

The error:

Some students use $3 \times 10^5 \ \text{m/s}$ (which is the speed of sound) or $3 \times 10^6 \ \text{m/s}$.

How to avoid:

Memorise the exact value for light in vacuum:

Write it at the top of your rough work before starting any calculation.

Mistake 3: Mishandling powers of 10 during division

The error:

When dividing $10^{25}$ by $10^8$, students sometimes subtract incorrectly (e.g., $25 - 8 = 17$ is correct, but they might write $10^{25} / 10^8 = 10^{33}$ by adding exponents).

How to avoid:

Use the law of exponents explicitly:

Write the step:

Mistake 4: Forgetting to account for the constant factor ($1/3$) when finding the order of magnitude

The error:

Students often say the order of magnitude is $10^{17}$ directly, without checking whether $\frac{1}{3} \times 10^{17}$ rounds to $10^{16}$ or $10^{17}$.

The rule (exam-critical):

Order of magnitude is the power of 10 when the number is written in scientific notation $a \times 10^n$, where $1 \le a < 10$.

• If $a < 3.16$ (i.e., $\sqrt{10}$), the order is $10^{n}$.
• If $a \ge 3.16$, the order is $10^{n+1}$.

Here:

Since $3.3 < 3.16$? No — $3.3 > 3.16$, so the order of magnitude is $10^{17}$ (not $10^{16}$).

How to avoid: …