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

In summary, a particle of mass m with velocity v1 transitions from one half-space with constant potential energy U1 to another with constant potential energy U2. The change in the direction of motion of the particle can be determined using the law of conservation of energy and the angles θ1 and θ2 between the normal to the plane and the velocities v1 and v2. The direction of force is vertical to the horizontal line and parallel to the plane of paper. Conservation of momentum holds along the horizontal axis, resulting in the equation v1sinθ1=v2sinθ2.
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jinjung
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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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  • #2
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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Related to Landau Vol.1 Mechanics(3rd ed.) Ch.II §7. Momentum Problem

1. What is the definition of momentum in Landau Vol.1 Mechanics?

The definition of momentum in Landau Vol.1 Mechanics is the product of an object's mass and velocity. It is a vector quantity and is represented by the symbol p.

2. How is momentum conserved in a closed system?

In a closed system, the total momentum of all objects remains constant. This means that the initial momentum before an interaction is equal to the final momentum after the interaction.

3. What is the difference between linear and angular momentum?

Linear momentum is the product of an object's mass and velocity in a straight line, while angular momentum is the product of an object's moment of inertia and its angular velocity around an axis.

4. How is the change in momentum related to force?

According to Newton's second law of motion, the change in momentum of an object is equal to the net force acting on the object multiplied by the time over which the force is applied. This can be expressed as p = FΔt.

5. Can momentum be negative?

Yes, momentum can be negative. This usually occurs when an object's velocity is in the opposite direction of its defined positive direction. The sign of momentum does not affect its conservation in a closed system.

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