Relativistic Mass (AQA A Level Physics): Revision Note

Equivalence of mass & energy

• Einstein showed in a 1905 paper that matter can be considered a form of energy, and hence, he proposed:

  • mass can be converted into energy

  • energy can be converted into mass

• This is known as mass-energy equivalence, and can be summarised by the equation:

$E=m{c}^2$

• Where:

  • $E$ = energy (J)

  • $m$ = mass (kg)

  • $c$ = the speed of light (m s^-1)

• Scenarios where mass is converted to energy or energy to mass include:

  • the fusion of hydrogen into helium in the centre of the Sun

  • the fission of uranium in nuclear power plants

  • nuclear weapons

  • high-energy particle collisions in particle accelerators

Relativistic mass

• An object's proper mass (that is, its mass measured by an observer at rest relative to the object) is also called its rest mass, $m_0$

• An observer in an inertial reference frame moving at speed $v$ relative to the object measures the object's mass as $m$, which is given by:

$m=\frac{m_0}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$

• This can also be written as:

$m=\gamma m_0$

• Because $\gamma >1$, when an object is moving relative to someone's frame of reference, they measure its mass as larger than its rest mass

  • The derivation for this relativistic mass is a good deal more complex than the derivation for relativistic time and length, and is not examinable, so it is omitted from this revision note

• One interesting outcome of this relativistic mass is the speed limit imposed on objects

  • As an object's speed nears the speed of light, its mass gets larger and larger

  • Very close to the speed of light, an object's mass tends towards an infinite mass

Relativistic energy

• Reconsidering Einstein's famous equation, taking relativistic effects into account:

$E=\left(\gamma m_0\right)c^2$

$E=\frac{m_0{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$

• The mass of a stationary object is its rest mass, $m_0$, so it has a rest energy of $E_0=m_0{c}^2$