Frequency of the em signal emitted by a rocket I `4xx10^(7)Hz`. If apparent frequency observed on earth is `3.2xx10^(7)Hz`, then velocity with which rocket is moving away is [speed of light = c]

Options

A. 0.5 c
B. 0.7 c
C. 0.9 c
D. 0.2 c

Answer

To find the velocity of the rocket moving away from the observer on Earth, we can use the Doppler effect for sound or electromagnetic waves. The formula for the apparent frequency when the source is moving away from the observer is given by:

\[ f' = f \left( \frac{c - v}{c} \right) \]

Where:
- \( f' \) is the apparent frequency observed on Earth,
- \( f \) is the emitted frequency of the rocket,
- \( c \) is the speed of light,
- \( v \) is the velocity of the rocket.

### Step 1: Identify the given values
- Emitted frequency, \( f = 4 \times 10^7 \, \text{Hz} \)
- Apparent frequency, \( f' = 3.2 \times 10^7 \, \text{Hz} \)
- Speed of light, \( c \) (we will keep it as \( c \) for now).

### Step 2: Substitute the values into the Doppler effect formula
We can rearrange the formula to solve for \( v \):

\[ f' = f \left( \frac{c - v}{c} \right) \]

Rearranging gives:

\[ \frac{f'}{f} = \frac{c - v}{c} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:

\[ f' \cdot c = f \cdot (c - v) \]

### Step 4: Expand and rearrange the equation
Expanding the right side:

\[ f' \cdot c = f \cdot c - f \cdot v \]

Rearranging to isolate \( v \):

\[ f \cdot v = f \cdot c - f' \cdot c \]

### Step 5: Factor out \( c \)
Factoring out \( c \):

\[ v = c \left( \frac{f - f'}{f} \right) \]

### Step 6: Substitute the values of \( f \) and \( f' \)
Now, substituting the values of \( f \) and \( f' \):

\[ v = c \left( \frac{4 \times 10^7 - 3.2 \times 10^7}{4 \times 10^7} \right) \]

Calculating the difference:

\[ v = c \left( \frac{0.8 \times 10^7}{4 \times 10^7} \right) \]

### Step 7: Simplify the fraction
This simplifies to:

\[ v = c \left( \frac{0.8}{4} \right) = c \left( \frac{1}{5} \right) \]

### Step 8: Final answer
Thus, the velocity of the rocket is:

\[ v = \frac{c}{5} \]

If we express this in terms of \( c \):

\[ v = 0.2c \]

### Conclusion
The velocity with which the rocket is moving away is \( 0.2c \).

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