State Hubble’s law.

Light from the galaxy M31 received on Earth shows a blue-shift corresponding to a fractional wavelength shift \(\frac{{\Delta \lambda }}{\lambda }\) of 0.001.

(i) Calculate the velocity of M31 relative to Earth.

(ii) The distance to M31 from Earth is 0.77 Mpc. Estimate, using the answer to (b)(i), a value of the Hubble constant.

(iii) Comment on your answer to (b)(ii).

recessional speed of (distant) galaxies is proportional to their separation / v = Hd; 
Symbols must be defined for the equation.

(i) \(\left( {\frac{{\Delta \lambda }}{\lambda } = \frac{v}{c} \Rightarrow } \right)v = 3 \times {10^5}\left( {{\rm{m}}{{\rm{s}}^{ - 1}}} \right)\) or 3×10²(kms^-1);
towards Earth;

(ii) 390 (kms^-1Mpc^-1); (allow other units: 390000(ms^-1Mpc^-1) , 0.39(ms^-1 pc^-1))

(iii) the value is wrong/cannot be used/cannot be relied upon / we cannot use this galaxy to calculate H;
Hubble’s law only applies to galaxies moving away / more distant galaxies;

The statement of Hubble's law is a frequent question and was generally well answered. However too many candidates still mention planets or stars rather than galaxies or omit to mention recessional velocity.

In (b)(i) the value of the velocity of M31 was almost always correct, but 'towards Earth' was usually missing. Hubble's constant calculations were done well in (b)(ii), with a variety of units used. It was expected that candidates would refer to the fact that M31 is not very distant and does not recede as a reason for the value in (b)(ii) being invalid. Few did. Instead reference was often made to the reason for the general uncertainty in the value of H - probably because the latter has been a more usual question in recent past papers. In teaching this topic it is obvious that a range of values for H can be found in textbooks. These can easily become out of date, but the value obtained in (b)(ii) was about 5 times that currently accepted. Candidates are obviously expected to have some idea of this value.