Energy-Momentum Relation (Edexcel A Level Physics)

• The equation for calculating the kinetic energy E_k of a particle m moving at velocity v is given by: 

• The formula for the momentum p of the same particle is:

• Combining these gives an equation that links kinetic energy to momentum, called the energy-momentum relation
  • Firstly, substituting the equation for velocity into the equation for kinetic energy gives:

• • Multiplying brackets out and simplifying gives: 

• Therefore the energy-momentum is presented as:

E_k = 

• Where:
  • E_k = kinetic energy (J)
  • p = momentum (kg m s^-1)
  • m = mass (kg)

• The energy-momentum relation is particularly useful for:
  • Calculations involving the kinetic energy of subatomic particles travelling at non-relativistic speeds (i.e. much slower than the speed of light)
  • Projectiles and collisions involving large masses 

Step 1: Write the energy-momentum relation

• • The energy-momentum relation is given by E_k = 

Step 2: Determine the mass of an alpha particle

• • An alpha particle is comprised of two protons and two neutrons
  • Therefore, the mass of an alpha particle m_α = 2m_p + 2m_n, where m_p and m_n is the mass of a proton and neutron respectively
  • So m_α = 2(1.67 × 10^–27) + 2(1.67 × 10^–27) = 6.68 × 10^–27 kg

Step 3: Substitute the momentum and the mass of the alpha particle into the energy-momentum relation

E_k = 

E_k = = 9.1 × 10^–13 J

Step 4: Convert the value of kinetic energy from J to MeV

1 MeV = 1.6 × 10^–13 J

• • Therefore:

9.1 × 10^–13 J = MeV = 5.7 MeV