Is it possible for a observer to move towards a stationary source emitting red light so that the red light appears to be green for the observer ? Wavelength of green is 5400Ả and that of red is 6200A. Velocity of light is `3 xx 10^(8)" m/s".`

Answer

To determine if an observer can move towards a stationary source emitting red light so that the red light appears green, we can use the Doppler effect for light. Here’s a step-by-step solution:

### Step 1: Understand the Wavelengths
We know the wavelengths of the colors involved:
- Wavelength of red light, \( \lambda_{red} = 6200 \, \text{Å} \)
- Wavelength of green light, \( \lambda_{green} = 5400 \, \text{Å} \)

### Step 2: Use the Doppler Effect Formula
The formula for the observed wavelength when the observer is moving towards a stationary source is given by:

\[ \lambda' = \lambda \left(1 - \frac{v}{c}\right) \]

Where:
- \( \lambda' \) is the observed wavelength,
- \( \lambda \) is the emitted wavelength,
- \( v \) is the speed of the observer,
- \( c \) is the speed of light.

### Step 3: Set Up the Equation
We want the observer to see the red light as green, so we set:

\[ \lambda' = \lambda_{green} = 5400 \, \text{Å} \]
\[ \lambda = \lambda_{red} = 6200 \, \text{Å} \]

Substituting these values into the Doppler effect formula:

\[ 5400 = 6200 \left(1 - \frac{v}{3 \times 10^8}\right) \]

### Step 4: Solve for \( v \)
Now we can solve for \( v \):

1. Rearranging the equation:

\[ 1 - \frac{v}{3 \times 10^8} = \frac{5400}{6200} \]

2. Calculate \( \frac{5400}{6200} \):

\[ \frac{5400}{6200} = 0.870967741935 \]

3. Now substituting back:

\[ 1 - \frac{v}{3 \times 10^8} = 0.870967741935 \]

4. Isolate \( \frac{v}{3 \times 10^8} \):

\[ \frac{v}{3 \times 10^8} = 1 - 0.870967741935 = 0.129032258065 \]

5. Multiply both sides by \( 3 \times 10^8 \):

\[ v = 0.129032258065 \times 3 \times 10^8 \]

6. Calculate \( v \):

\[ v \approx 3.87096774195 \times 10^7 \, \text{m/s} \approx 4.4 \times 10^7 \, \text{m/s} \]

### Final Answer
The speed of the observer should be approximately \( 4.4 \times 10^7 \, \text{m/s} \) for the red light to appear green.

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