Introduction:
In this scenario, we have two particles, A and B, where particle A is moving with an initial direction of motion and particle B is stationary. When these particles collide, they experience a perfectly elastic collision, meaning that both momentum and kinetic energy are conserved. The particles fly apart symmetrically relative to the initial direction of motion of particle A with an angle of divergence θ.

Understanding Elastic Collision:
Elastic collisions are characterized by the conservation of both momentum and kinetic energy. This means that the total momentum before the collision is equal to the total momentum after the collision, and the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Analysis of the Collision:
1. Initial momentum:
- Particle A has an initial momentum, which is the product of its mass (m_A) and velocity (v_A).
- Particle B is stationary, so it has zero initial momentum.

2. Final momentum:
- After the collision, the particles fly apart symmetrically with an angle of divergence θ. Therefore, the magnitude of their momenta will be equal.
- Let the final momentum of both particles be P.

3. Conservation of momentum:
- According to the law of conservation of momentum, the total initial momentum is equal to the total final momentum.
- So, m_Av_A + 0 = P + P
- m_Av_A = 2P

4. Conservation of kinetic energy:
- In a perfectly elastic collision, the total initial kinetic energy is equal to the total final kinetic energy.
- The initial kinetic energy is given by (1/2)m_Av_A².
- The final kinetic energy of both particles will be equal since they have the same magnitude of momentum.
- Therefore, (1/2)m_Av_A² = (1/2)P² + (1/2)P² = P²

5. Solving for the ratio of masses:
- Dividing the equation m_Av_A = 2P by the equation (1/2)m_Av_A² = P², we get:
(m_Av_A) / [(1/2)m_Av_A²] = (2P) / P²
- Simplifying the left-hand side, we obtain:
2 / v_A = 2 / P
- Canceling out the common terms, we get:
v_A = P
- Substituting this result back into the equation m_Av_A = 2P, we find:
m_AP = 2P
- Div