A galaxy is moving away from the earth at a speed $286 \mathrm{~km/s}$. The shift in wavelength of a redline at 630 nm is $x \times 10^{-10} \mathrm{~m}$. The value of $x$, to the nearest integer, is ____ [Take the value of speed of light c, as $3 \times 10^{8} \mathrm{~ms}^{-1}$]

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Waves & Sound

A galaxy is moving away from the earth at a speed

286 km/s

. The shift in wavelength of a redline at 630 nm is

x \times 1 0^{- 10} m

. The value of

x

, to the nearest integer, is ____
[Take the value of speed of light c, as

3 \times 1 0^{8} ms^{- 1}

]

Answer: 6

Using the Doppler shift formula for light:

\frac{Δ λ}{λ} = \frac{v}{c}

. Given

v = 286 \times 1 0^{3}

m/s,

λ = 630 \times 1 0^{- 9}

m, and

c = 3 \times 1 0^{8}

m/s. We have

Δ λ = λ \times \frac{v}{c} = 630 \times 1 0^{- 9} \times \frac{286 \times 1 0 ^{3}}{3 \times 1 0 ^{8}} = 630 \times 1 0^{- 9} \times \frac{286}{3 \times 1 0 ^{5}} = 630 \times 1 0^{- 9} \times 0.000953 \approx 6 \times 1 0^{- 10}

m. Thus

x = 6

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Integer answer