A ball is attached to one end of a wire, the other end beingfastened to the ceil

Question

A ball is attached to one end of a wire, the other end beingfastened to the ceiling. The wire is held horizontal, and the ballis released from rest (see the drawing). It swings downward andstrikes a block initially at rest on a horizontal frictionlesssurface. Air resistance is negligible, and the collision iselastic. The masses of the ball and block are, respectively, 1.6 kgand 2.3 kg, and the length of the wire is 1.20 m. Find the velocity(magnitude and direction) of the ball (a) justbefore the collision, and (b) just after thecollision.

Explanation / Answer

   as the ball swings downward, its speedincreases. Therefore, its momentum and kinetic energy, which dependon the    ball’s speed, also increase    thus, these quantities are not conserved(they do not remain constant) during the motion    since air resistance is negligible, however,the work done by nonconservative forces is zero    Wnc = 0 J    so the total mechanical energy, which is thesum of the kinetic and potential energies, is conserved    during the collision, the xcomponent of the total momentum of the ball/block system and thetotal kinetic energy of the    system are conserved    the horizontal component of the totalmomentum is conserved, because the horizontal surface isfrictionless and so the net    average external force acting in thehorizontal direction is zero    the total kinetic energy is conserved, sincethe collision is known to be elastic (a)    as the ball falls, the total mechanicalenergy is conserved    thus, the total mechanical energy at the topof the swing is equal to that at the bottom    mball gho=(1 /2)mball vf2    expression ho is the initialheight of the ball above its level at the bottom of the swing,which is where h = 0    solving for the speed vfof the ball just before the collision gives    vf = 2 gho    at the bottom of the swing the ball ismoving horizontally and to the right, which we take to be the+x direction    thus the velocity of the ball is (b)    the conservation of linear momentum and theconservation of the total kinetic energy can be used to describethe behavior    of the system during the collision    vf1 = [(m1 -m2) / (m1 + m2)vo1    wherevf1 is the final velocity ofthe ball    m1 and m2are, respectively, the masses of the ball and block, andv01 is the velocity of the ball just before the    collsionince    vf= .......m/swe findthat    vf1 = ............ m / s    the minus sign indicates that the ballrebounds to the left after the collision.

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