# Energy-Momentum Relation(Edexcel A Level Physics)

Author

Katie M

Expertise

Physics

## Deriving the Energy-Momentum Relation

• The equation for calculating the kinetic energy Ek 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:

Ek

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

#### Exam Tip

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

Step 1: Write the energy-momentum relation

• The energy-momentum relation is given by Ek

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

Ek= 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

#### Exam Tip

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) × 10= 1.60 × 10-13

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