A pi meson of mass m moves with a speed u in the positive y direction in frame S with parameters BETA = u/c and
GAMMA = 1/[sqrt(1 - BETA^2)].
The pion decays into two gamma rays. An observer S' moves in the positive x direction with speed v = BETA'c and detects the two photons.
A. Compute the sum of the x components of momentum in S' for the photons.
B. Compute the combined energy of the two photons in S'.
To solve this problem, we will use the principles of special relativity to relate the quantities in frame S to those in frame S'. Let's break down the problem into smaller steps:
Step 1: Find the velocities of the two particles in frame S'
We know the velocity of the pi meson in frame S is u in the positive y direction. To find the velocities of the two gamma rays in frame S', we need to apply the velocity addition formula in special relativity.
The velocity addition formula states that the velocity v' of an object in frame S' can be calculated as:
v' = (v - u) / (1 - v * u / c^2)
In this case, v is the velocity of frame S' with respect to frame S, which is given as v = BETA'c. Plugging in the values, we have:
v' = (BETA'c - u) / (1 - BETA' * u / c^2)
Step 2: Calculate the momenta of the photons in frame S'
The momentum p' of an object in frame S' can be calculated as:
p' = (m * v') / sqrt(1 - v'^2 / c^2)
In this case, m is the mass of the gamma rays, which can be assumed to be zero because photons are massless particles.
Step 3: Find the sum of the x-components of momentum in frame S'
To find the sum of the x-components of momentum in frame S', we need to add up the x-components of momentum for the two gamma rays.
Let's denote the x-component of momentum for the first photon as p'x1, and for the second photon as p'x2. The sum of the x-components of momentum, Σp'x, can be calculated as:
Σp'x = p'x1 + p'x2
Step 4: Calculate the combined energy of the two photons in frame S'
The energy E' of an object in frame S' can be calculated as:
E' = γ * mc^2
In this case, m is again assumed to be zero because photons are massless particles. γ is the Lorentz factor, which can be calculated as γ = 1 / sqrt(1 - v'^2 / c^2).
Now that we have outlined the steps, we can proceed with the calculations.