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Edexcel A Level Physics:復習筆記6.6 Energy-Momentum Relation

Category: A-level課程, 教材筆記, 福利干貨 Date: 2022年10月11日上午9:42

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 velocityinto the equation for kinetic energy gives:

- Multiplying brackets out and simplifying gives:

Therefore the?energy-momentum?is presented as:

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

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

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

1 MeV = 1.6 × 10–13?J

- Therefore:

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) × 106?= 1.60 × 10-13

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