Final answer:
A particle's velocity and rebound angle after colliding with a wall are determined by analyzing the components of the initial velocity, using the coefficient of restitution for the perpendicular component, and applying trigonometry and the Pythagorean theorem to find the final velocity magnitude and direction.
Step-by-step explanation:
When a particle with mass m strikes a smooth wall at speed u and an angle θ from the normal, we must consider both the perpendicular and parallel components of the velocity to find the velocity and rebound angle of the particle after the collision. Since the wall is smooth, there is no friction, and hence the parallel component of the velocity remains unchanged. The perpendicular component, however, is affected by the coefficient of restitution E, which is a measure of how bouncy the collision is.
The initial velocity can be split into two components: ux = ucos(θ) parallel to the wall, and uy = usin(θ) perpendicular to the wall. After the collision, the parallel component remains ux = ucos(θ), while the perpendicular component becomes Euy = Eusin(θ) but in the opposite direction. Using trigonometry, we can find the rebound angle θ' by taking the arctangent of the ratio of the perpendicular (post-collision) and parallel velocity components θ' = arctan(Euy/ux).
The final velocity magnitude v' can be found using the Pythagorean theorem on the components: v' = sqrt(ux2 + (Euy)2). The direction of the velocity can also be expressed as the rebound angle θ'.