Velocity Changes and Collisions: Exploring the Dynamics of a Particle in a Moving Box
a) What are the particle's velocities before and after colliding with the moving wall in the frame of the moving wall?
1) v 1: v_wall
2) v 2: -v_wall
b) What is the change in velocity of the particle during the elastic collision?
Δv = v2 - v1 = -2 * v_wall
c) What is the particle's velocity after the collision in the lab frame?
v2 = -v0
d) What is the velocity of the particle after colliding with the wall at rest in the lab frame?
v3: ?
e) What is the average acceleration of the particle over a cycle of collisions?
a: ?
e) How do the total energy and the quantity L^2E of the particle relate in quantum mechanics?
L^2E: ?
Answers:
a) In the frame of the moving wall, the particle's velocity before the collision (v1) is equal to the velocity of the wall (v_wall), and the particle's velocity after the collision (v2) is equal to the negative of its velocity before the collision.
b) The change in velocity of the particle (Δv = v2 - v1) over the elastic collision is equal to -2 times the velocity of the wall.
c) In the lab frame, where v1 = v0, the velocity of the particle after the collision (v2) is equal to -v0.
When analyzing the motion of a particle in a moving box, it is essential to consider the changes in velocity during collisions. In the frame of the moving wall, the particle's velocity before the collision is equal to the velocity of the wall (v_wall). After the collision, the particle's velocity changes direction but preserves its magnitude, becoming equal to the negative of its velocity before the collision. This change in velocity can be calculated as -2 times the velocity of the wall.
Transitioning to the lab frame, where the initial velocity of the particle is v0, the velocity of the particle after the collision is also in the opposite direction but maintains the same magnitude as its initial velocity (-v0). This rotation of velocity showcases the dynamics of collisions in different reference frames.