Relativity

Step 1: Understanding the relativistic velocity addition formula.
In relativity, the velocity of the particle as observed from can be obtained using the relativistic velocity addition formula: Here, and are the speeds of the particle and the frame , respectively. Substituting these values gives the speed of the particle as seen from as .
Step 2: Conclusion.
Thus, the correct answer is option (C) because the speed is .

Step 1: Understanding the relativistic velocity addition formula.
In relativity, the velocity of the particle as observed from can be obtained using the relativistic velocity addition formula: Here, and are the speeds of the particle and the frame , respectively. Substituting these values gives the speed of the particle as seen from as .
Step 2: Conclusion.
Thus, the correct answer is option (A) because the speed is .

Step 1: Use Lorentz contraction.
Length in frame: where is relative velocity between the rods.
Given: m, m.

Step 2: Solve for relative velocity .




.

Step 3: Relate to .
Using relativistic velocity addition: .

Step 4: Solve for .
.
Solving gives .

Step 1: Use Lorentz time transformation.
, where .

Step 2: Evaluate parameters.
, hence .

Step 3: Compute transformed times.
Event : → .
Event : → .

Step 4: Substitute values.
.
.

Step 5: Take magnitude (time difference).
after correcting unit conversion.

Step 1: Analyze conservation laws.
In any collision process, the fundamental conserved quantities are total momentum and total energy (including rest and kinetic energies).

Step 2: Relativistic collision condition.
When two particles of equal and opposite momenta collide and stick together, total momentum before collision is zero. Hence, after collision, total momentum remains zero — momentum conserved.

Step 3: Check kinetic and rest mass conservation.
Total kinetic energy is converted to other energy forms (rest mass, internal energy, etc.), so it is not conserved. However, total energy (including rest energy) is conserved. Rest mass, on the other hand, changes due to binding energy.

Step 4: Conclusion.
Hence, total momentum and total energy are conserved.

Step 1: Understanding the relativistic energy transformation.
The energy of a particle in relativity is given by the formula where is the Lorentz factor, and is the velocity of the particle.
Step 2: Energy in the initial frame .
For the frame , the particle is moving with a velocity , so the Lorentz factor for frame is Thus, the energy in frame is Step 3: Energy in the frame .
In frame , the particle is moving with a velocity relative to , and itself is moving at with respect to . The relative velocity between the particle and the new frame is given by the relativistic velocity addition formula: Thus, , so the energy in the frame would be the same as in the original frame. Hence, Step 4: Final Answer.
The ratio .

Step 1: Apply the Lorentz transformation for time.
Given for all three events, this simplifies to: Step 2: Substitute for each event.
For
For
For
Step 3: Compare the times.
Step 4: Final Answer.
Hence, the correct order is after interpreting the negative direction.

Step 1: Understanding the Concept:
This problem involves an inelastic collision between two relativistic particles. In any collision, both total energy and total momentum are conserved. We can use the conservation laws and the relativistic energy-momentum relation to find the rest mass of the final particle.
Step 2: Key Formula or Approach:
The key principles are the conservation of relativistic energy and momentum.
Relativistic energy:
Relativistic momentum:
Conservation of Energy:
Conservation of Momentum:
A very useful tool is the energy-momentum invariant relation for a system: , where is the total rest mass of the system. For a single particle, this is its rest mass M.
Step 3: Detailed Explanation:
Let's denote the Lorentz factor as .
Initial State (before collision):
Particle 1: rest mass , velocity .
Particle 2: rest mass , velocity .
Total initial energy ( ):
Total initial momentum ( ):
Final State (after collision):
The two particles coalesce into a single particle of rest mass M. Let its velocity be V.
Final energy and momentum of this single particle are related by .
Applying Conservation Laws:
By the principle of conservation, the total energy and momentum of the system are conserved.


Now we use the energy-momentum invariant for the final particle:
Substituting the values of and :
Since , we have . Substituting this:
Cancelling from both sides:
Step 4: Final Answer:
The rest mass M of the resulting particle is . This corresponds to option (A).