Problem set - p6

Question

1. Find the speed and momentum of a proton whose total energy is 3.500 GeV. Hints: 1 Convert to joules first; ii. What is conversion from GeV to eV

Answer 1

To calculate the speed and momentum of a proton with a total energy of 3.500 GeV, we need to convert the energy from GeV to joules and then apply the appropriate formulas.

1. Conversion from GeV to eV:

1 GeV (giga electronvolt) = 1 × 10^9 eV (electronvolts)

So, 3.500 GeV = 3.500 × 10^9 eV

2. Conversion from eV to joules:

1 eV = 1.602 × 10^-19 J (joules)

To convert 3.500 × 10^9 eV to joules, we multiply it by the conversion factor:

3.500 × 10^9 eV × (1.602 × 10^-19 J/eV) ≈ 5.607 × 10^-10 J

Now that we have the energy in joules, we can proceed to calculate the speed and momentum.

The total energy of a particle can be expressed as:

E = (γ - 1)mc^2

Where:

E = total energy (in joules)

γ = Lorentz factor (γ = 1 / √(1 - (v^2 / c^2)))

m = rest mass of the proton (approximately 1.67 × 10^-27 kg)

c = speed of light in a vacuum (approximately 3 × 10^8 m/s)

Since we are given the total energy, we can rearrange the formula to solve for the velocity (v):

v = c √(1 - (1 / (γ^2)))

To calculate the momentum (p), we can use the formula:

p = γmv

Let's plug in the values:

E = 5.607 × 10^-10 J

m = 1.67 × 10^-27 kg

c = 3 × 10^8 m/s

First, we calculate γ:

γ = √(1 + (E / (mc^2)))

Next, we calculate v:

v = c √(1 - (1 / (γ^2)))

Finally, we calculate p:

p = γmv

Calculating the exact values requires more precision, but this should give you a general idea of the process.