Landau Vol.1 Mechanics(3rd ed.) Ch.II §7. Momentum Problem

[jinjung]

Homework Statement

A particle of mass m moving with velocity v₁ leaves a half-space in which its porential energy is a constant U₁ and enters another in which its potential energy is a different constant U₂.
Determine the change in the direction of motion of the particle.

Homework Equations

The porential energy is independent of the co-ordinates whose axes are parallel to the plane separating the half-spaces. The component of momentum in that plane is therefore conserverd. Denoting by θ₁ and θ₂ the angles between the normal to the plane and the velocities v₁ and v₂ of the particle before and after passing the plane, we have v₁sinθ₁=v₂sinθ₂. The relation between v₁ and v₂ is given by the law of conservation of energy, and the result is

The Attempt at a Solution

I have no idea what a situation of the problem is.

Is the situation of the problem this?
If so, is a direction of force vertical to the paper?

And why v₁sinθ₁=v₂sinθ₂?

 

[Delta2]

The situation is almost as your image, it is just that the angles denoted as $\theta_i$ by the book are the complementary angles of the angles you have draw in the image.(because the book says angles formed by the normal to the plane and the velocity, instead you have draw angles formed by the parallel to the plane and the velocities.)

The direction of the force is vertical to the horizontal line (why?) and parallel the plane of paper. It is not vertical to the paper. Because we have no horizontal forces involved, conservation of momentum holds along the horizontal axis so what do you get if you apply conservation of momentum along the horizontal axis ?(remember to first redraw the angles $\theta_i$).