Animated Solution for Physics - Waves: A galaxy is moving away from the Earth at a speed of $286{\text{ kms}}^{-1}$. The shift in the wavelength of a red line at $630\text{ nm}$ is $x\times 10^{-10}\text{ m}$. The value of $x$ to the nearest integer, is ........... . (Take the value of speed of light $c$, as $3\times 10^8{\text{ ms}}^{-1}$)

Imagine you are standing on Earth, looking up at the night sky. A distant galaxy is moving away from us at a staggering speed of $286\text{ km/s}$. Because it is moving away, the light it emits gets stretched out as it travels through space towards us. This phenomenon is called redshift, which is a classic example of the Doppler effect for light.

To work with standard SI units, we first convert the given values. The velocity of the galaxy is $v=286\times 10^3\text{ m/s}$. The original wavelength of the red line is $\lambda =630\text{ nm}=630\times 10^{-9}\text{ m}$.

To find out exactly how much the wavelength stretches, we use the Doppler shift formula for light. When the velocity of the source is much less than the speed of light ($v\ll c$), the fractional change in wavelength is equal to the ratio of the source's velocity to the speed of light:

Here, $\Delta \lambda$ represents the shift in wavelength, and $c=3\times 10^8\text{ m/s}$ is the speed of light in a vacuum. This is a fundamental relation used extensively in astrophysics to determine how fast galaxies are receding from us.

Let's rearrange the formula to solve for $\Delta \lambda$ and carefully substitute our known values:

To avoid silly mistakes, it is always best to group the numbers and the powers of ten separately. We can easily divide $630$ by $3$ to get $210$:

Now, combining the powers of ten ($3-9-8=-14$) and multiplying the integers ($286\times 210=60060$):

The question asks for the answer in the format of $x\times 10^{-10}\text{ m}$. To achieve this, we shift the decimal point four places to the left, which multiplies the coefficient by $10^4$:

Comparing our result with the given expression $x\times 10^{-10}\text{ m}$, we see that $x=6.006$. Rounding to the nearest integer, we get $x=6$.

Always remember to pay attention to the direction of motion. If the galaxy were moving towards us, the light waves would be compressed, resulting in a shorter wavelength. This is known as a blueshift.