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

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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
Solution.png


The Attempt at a Solution


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

20180719_161257.jpg

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

And why v1sinθ1=v2sinθ2?
 

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Delta2
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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##).
 
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