What happens to a particle in a 1D box when one of the walls starts to move very slowly, changing the size of the box from a constant length L to L(t)? Specifically, what are the particle's velocities before and after colliding elastically with the moving wall?
In the scenario described, when a particle in a 1D box of length L experiences a change in box size due to one of the walls moving very slowly, the particle's velocities undergo specific changes. Before colliding elastically with the moving wall, the particle's velocity (v1) remains equal to its initial velocity (v0). However, after the elastic collision, the particle's velocity (v2) changes direction but preserves its magnitude.
To understand this concept more clearly, let's delve into the physics behind the particle's velocities before and after the collision.
Conservation of Momentum in Elastic Collision
In the frame of the moving wall, the particle's velocity before colliding elastically with the wall is v1 (which is the same as its initial velocity v0). After the collision, the particle's velocity changes direction, resulting in a new velocity v2.
To determine v1 and v2, we need to consider the conservation of momentum in an elastic collision. Before the collision, the particle's momentum is represented by p1 = mv0. After the collision, although the direction of velocity changes, the magnitude of momentum remains the same, leading to a new momentum of p2 = -mv2. By equating the magnitudes of these momenta, we can solve for v2:
mv0 = mv2
v2 = -v0
Therefore, the particle's velocity before the collision (v1) remains v0, while its velocity after the collision (v2) changes direction to -v0.
Understanding the dynamics of elastic collisions and the conservation of momentum is crucial in predicting the behaviors of particles in different scenarios, such as the one described in the problem. By analyzing the changes in velocity before and after collisions, physicists can gain insights into the interactions between particles and their environments.
In summary, the particle's velocity before colliding elastically with the moving wall remains v1 = v0, and its velocity after the collision changes to v2 = -v0. This understanding highlights the fundamental principles of momentum conservation and the directional changes that particles undergo during collisions in a dynamic system like a 1D box with moving walls.