国际物理林匹克竞赛试题ThQIMechanicsSolution.docVIP

Mechanics – Problem I (8 points) A particle moves along the positive axis (one-dimensional situation) under a force having a projection on , as represented, as function of , in the figure 1.1. In the origin of the axis is placed a perfectly reflecting wall. A friction force, with a constant modulus, acts everywhere on the particle. The particle starts from the point having the kinetic energy. a. Find the length of the path of the particle until its’ final stop b. Plot the potential energy of the particle in the force field. c. Qualitatively plot the dependence of the particle’s speed as function of its’ coordinate. Figure 1.1 Problem I – Solution a. It is possible to make a model of the situation in the problem, considering the Ox axis vertically oriented having the wall in its’ lower part. The conservative force could be the weight of the particle. One may present the motion of the particle as the vertical motion of a small elastic ball elastically colliding with the ground and moving with constant friction through the medium. The friction force is smaller than the weight. The potential energy of the particle can be represented in analogy to the gravitational potential energy of the ball,, considering . As is very well known, in the field of a conservative force, the variation of the potential energy depends only on the initial and final positions of the particle, being independent of the path between those positions. For the situation in the problem, when the particle moves towards the wall, the force acting on it is directed towards the wall and has the modulus l ( 1.1) ( 1.2) As a consequence, the motion of the particle towards the wall is a motion with a constant acceleration having the modulus ( 1.3) During the motion, the speed of the particle increases. Hitting the wall, the particle starts moving in opposite direction with a speed equal in modulus with the one it had before the collision. When the particle moves away