If we intercept an electron having total energy 1533 MeV that came from Vega, which is 26 ly from us, how far in light-years was the trip in the rest frame of the electron? (Answer: 8.7*10^-3 ly)
rest energy of electron m₀c²=0.512 MeV
E=mc²=1533 MeV
mc² =m₀c²/sqrt(1-β²)... (1)
τ=τ₀/sqrt(1-β²) …..(2)
Divide (1) by (2)
mc²/τ= m₀c²/τ₀
τ₀=τ• m₀c²/mc²= 26•0.512/1533=0.00868 ly
Thank you so much!
To determine the distance traveled by the electron in its rest frame, we need to use two important concepts from special relativity: time dilation and length contraction.
Firstly, let's consider time dilation. In the rest frame of an object, time passes more slowly compared to an observer in a different frame moving relative to the object. The time dilation formula is given by:
t' = t / γ
where t is the time measured in the observer's frame, t' is the time measured in the rest frame of the object, and γ (gamma) is the Lorentz factor:
γ = 1 / √(1 - v^2/c^2)
In this case, the observer is located on Earth and the electron is moving at a relativistic speed. We want to find the time experienced by the moving electron, so we need to rearrange the time dilation equation:
t' = t / γ --> t = t' * γ
Notice that t is the time as measured by the observer on Earth, and t' is the time experienced by the electron.
Now, let's consider length contraction. Length contraction is the phenomenon where the length of an object in motion appears shorter in the direction of its motion as measured by an observer in a different frame. The length contraction formula is given by:
L' = L * √(1 - v^2/c^2)
where L is the length measured in the observer's frame, L' is the length measured in the rest frame of the object, and v is the velocity of the moving object.
Rearranging the length contraction equation, we get:
L = L' / √(1 - v^2/c^2)
In this case, we want to find the distance traveled by the electron. Since the electron is moving close to the speed of light, we need to take into account length contraction.
Now, let's calculate the values required to solve the problem:
Total energy of the electron, E = 1533 MeV
Velocity of the electron, v = c (speed of light)
Distance to Vega, d = 26 light-years
First, convert the energy of the electron into its relativistic momentum using the Einstein mass-energy equivalence formula:
E = √(m^2c^4 + p^2c^2)
Since the electron is ultra-relativistic (moving at a speed close to the speed of light), we can ignore the mass term (m) and solve for the momentum (p):
p = √(E^2 / c^2)
Given the total energy E in MeV, we have:
p = √((E * 10^6)^2 / c^2) [convert MeV to eV]
Substitute the values into the equation and calculate p.
Next, calculate the time experienced by the electron in its rest frame using the time dilation equation:
t' = d / v
Substitute the values into the equation and calculate t'.
Finally, compute the distance traveled by the electron in its rest frame using length contraction:
L = d * √(1 - v^2/c^2)
Substitute the values into the equation and calculate L.
The result will give you the distance traveled by the electron in light-years in its rest frame, which is approximately 8.7 * 10^-3 ly.