Two starships, the Enterprise and the Constitution, are approaching each other head-on from a great distance. The separation between them is decreasing at a rate of 713.0 km/s. The Enterprise sends a laser signal toward the Constitution. If the Constitution observes a wavelength ? = 712.2 nm, what wavelength was emitted by the Enterprise? Express your answer with four significant digits.

Explanation:

Assume the enterprise is sitting with zero velocity, while the Constitution is moving towards the enterprise at a speed of 713km/s

V = 713,000 m/s

Since the constitution is moving into the wave, then it will be compressed.

So, we want to find the original wavelength λ

So using relativity formula

λ' = λ(c+Vo)/(c + V)

Given that,

λ' = 712.2nm = 712.2 × 10^-9m

c = 3×10^8 m/s

Vo = 0

V = 713,000 m/s

Then,

λ' = λ(c+Vo)/(c + V)

712.2 ×10^-9 = λ (3×10^8+0) / (3×10^8+713,000)

712.2×10^-9=λ(3×10^8)/(3.00713×10^8)

712.2×10^-9=λ × 0.9976

λ = 712.2 × 10^-9 / 0.9976

λ = 713.9 × 10^-9

λ = 713.9 nm

Which is a bit longer as expected

3.2x10^7