Elastic And Inelastic Collisions

Step 1: Understand the System - A particle of mass collides elastically with a particle of mass initially at rest. - We seek the maximum angular deviation of the heavier particle (mass ) after the collision.
Step 2: Apply Conservation Laws - We use conservation of momentum (x and y) and conservation of kinetic energy (elastic collision).
Momentum Conservation: x-direction: y-direction: Kinetic Energy Conservation:
Step 3: Relate Variables and Find Maximum Angle - The goal is to express in terms of known constants and maximize it. This is best done with as independent variables and related to . - Solve Eqn. (2) for : - Solve Eqn. (1) for : - Use : - Use Eqn. (3) to eliminate : - Solve the quadratic for using the quadratic formula: - For to be real, we require: - Therefore: - So the maximum scattering angle is .
Alternate Solution using the Approximation: In an elastic collision, the maximum scattering angle of the heavier particle (mass ) when colliding with a lighter particle (mass ) is given by: This formula is valid when . In this case, and . Thus: Step 4: Conclusion The maximum angular deviation of the heavier particle is . Final Answer: The correct option is:

Assume: - Mass of incoming particle: - Target mass at rest: - Elastic collision: momentum and kinetic energy conserved Use result from elastic collision (scattering in lab frame): If a moving particle of mass collides elastically with a stationary particle of mass , and both particles move after collision, then the angle between their velocity vectors satisfies: Final Answer