A photon of energy E collides with a stationary particle of rest mass m0 and is absorbed by it.
what is the velocity of the resulting composite particle ?
what is the mass of the resulting composite particle?
(use conservation of momentum and energy)
answer to part a is v=E/(E+mc^2), not sure about part b.
To find the velocity of the resulting composite particle, we can use the conservation of momentum.
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Before the collision:
The photon has momentum p1 = E/c, where c is the speed of light.
The stationary particle has momentum p2 = 0 since it is stationary.
After the collision:
The resulting composite particle has momentum p3 = mv3, where v3 is the velocity of the composite particle.
The photon no longer exists as a separate particle.
Using the conservation of momentum equation: p1 + p2 = p3, we can write:
E/c + 0 = mv3
Rearranging the equation, we get:
v3 = E/(mc)
Now, let's find the mass of the resulting composite particle using the conservation of energy.
According to the conservation of energy, the total energy before the collision is equal to the total energy after the collision.
Before the collision:
The photon has energy E.
The stationary particle has rest energy E0 = m0c^2.
After the collision:
The resulting composite particle has energy E3 = mc^2, where m is the mass of the composite particle.
Using the conservation of energy equation: E + E0 = E3, we can write:
E + m0c^2 = mc^2
Rearranging the equation, we get:
m = (E + m0c^2)/c^2
So, the velocity of the resulting composite particle is v3 = E/(mc), and the mass of the resulting composite particle is m = (E + m0c^2)/c^2.
To determine the velocity and mass of the resulting composite particle, we can make use of the principles of conservation of momentum and energy.
1. Conservation of momentum:
We know that momentum is conserved in any isolated system. In this case, since the photon collides with a stationary particle, the initial momentum of the system is given by the momentum of the photon, p_initial = E/c, where E is the energy of the photon and c is the speed of light.
After the collision, the resulting composite particle will have a certain velocity v. The final momentum of the system is then given by p_final = (m_total+m0)v, where m_total is the total mass of the system after the collision (including both the stationary particle and the resulting composite particle).
Applying conservation of momentum, we equate p_initial to p_final: E/c = (m_total+m0)v.
2. Conservation of energy:
Energy is also conserved in an isolated system. The total initial energy of the system is E, which is the energy of the photon. After the collision, the resulting composite particle will possess kinetic energy, given by (1/2)(m_total+m0)v^2, as it gains velocity v. Therefore, the total final energy of the system is given by (1/2)(m_total+m0)v^2.
Applying conservation of energy, we equate the initial energy E to the final energy: E = (1/2)(m_total+m0)v^2.
Now, we have a system of equations:
E/c = (m_total+m0)v -- Equation (1)
E = (1/2)(m_total+m0)v^2 -- Equation (2)
We can solve these two equations simultaneously to find the values of v and m_total.
Solving these equations involves some algebraic manipulation. Rearranging Equation (1) for v, we get:
v = E/(c(m_total+m0))
Substituting this value of v into Equation (2), we get:
E = (1/2)(m_total+m0)(E/(c(m_total+m0)))^2
Simplifying the equation, we find:
1 = (1/2)(E^2/(c^2(m_total+m0)^2))
Solving for m_total, we get:
m_total = (E^2)/(c^2(E^2-(m0c^2)^2))
Finally, substituting the value of m_total into Equation (1), we can solve for v:
v = E/(c(m_total+m0))
Note: It's important to convert units appropriately when necessary, and ensure that the equations are dimensionally consistent.