A LevelPhysicsEdexcelRevision Notes6. Further MechanicsMomentum & ImpulseEnergy-Momentum Relation

Energy-Momentum Relation (Edexcel A Level Physics): Revision Note

Exam code: 9PH0

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Updated on

Deriving the Energy-Momentum Relation

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

E subscript k equals 1 half m v squared

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

p space equals space m v

  • Combining these gives an equation that links kinetic energy to momentum, called the energy-momentum relation

    • Firstly, substituting the equation for velocity v equals p over minto the equation for kinetic energy gives:

    E subscript k equals 1 half m open parentheses p over m close parentheses squared

    • Multiplying brackets out and simplifying gives: 

E subscript k equals 1 half m p squared over m squared equals 1 half p squared over m

  • Therefore the energy-momentum is presented as:

Ek = fraction numerator p squared space over denominator 2 m end fraction

  • Where:

    • Ek = kinetic energy (J)

    • p = momentum (kg m s-1)

    • m = mass (kg)

Examiner Tips and Tricks

This is a common derivation, so make sure you're comfortable with deriving this from scratch! Think carefully about the algebra on each step.

Using the Energy-Momentum Relation

  • 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 

Worked Example

Calculate the kinetic energy, in MeV, of an alpha particle which has a momentum of 1.1 × 10–19 kg m s–1.

Use the following data:

  • Mass of a proton = 1.67 × 10–27 kg

  • Mass of a neutron = 1.67 × 10–27 kg

Answer:

Step 1: Write the energy-momentum relation

  • The energy-momentum relation is given by Ek = fraction numerator p squared over denominator 2 m end fraction

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 mα = 2mp + 2mn, where mp and mn is the mass of a proton and neutron respectively

  • So mα = 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

Ek = fraction numerator p squared over denominator 2 m end fraction

Ek = fraction numerator left parenthesis 1.1 cross times 10 to the power of negative 19 end exponent right parenthesis squared over denominator 2 cross times left parenthesis 6.68 cross times 10 to the power of negative 27 end exponent right parenthesis end fraction= 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 = fraction numerator 9.1 cross times 10 to the power of negative 13 end exponent over denominator 1.6 cross times 10 to the power of negative 13 end exponent end fractionMeV = 5.7 MeV

Examiner Tips and Tricks

Calculations with the energy-momentum equation often require changing units, especially between eV and J due to it commonly being used for particles. Remember that 1 eV = 1.60 × 10-19 J. Therefore

eV → J = × (1.60 × 10-19)

J → eV = ÷ (1.60 × 10-19)

The prefix 'mega' (M) means × 106 therefore, 1 MeV = (1.60 × 10-19) × 106 = 1.60 × 10-13

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Author
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Katie M

Author: Katie M

Expertise: Curriculum Expert

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

Caroline Carroll

Reviewer: Caroline Carroll

Expertise: Head of Content Delivery

Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about delivering high-quality resources to help students achieve their full potential.