A small ball of mass m is projected with a minimum horizontal velocity v_(0) on a smooth wedge of mass M so that it will reach the highest point of the wedge. Find the value of v_(0)

A small ball of mass m is projected with a minimum horizontal velocity

1.	A small ball of mass m is projected with a minimum horizontal velocity v_(0) on a smooth wedge of mass M so that it will reach the highest point of the wedge. Find the value of v_(0)
Answer» <P> Solution : if the ball reaches at point `P`, the VELOCITY of the ball with respect to wedge should be `sqrt(gR)` using work energy theorem from centre of mass frame at `A` and the HIGHEST point `P` ltbr `W_(ext)+W_(int)+0=(/_\K)-(cm)` `W_("gravity")=(/_\K)_(CM)` `-mg(2R)=[1/2muv_(rel)^(2)]_("final")-[1/2muv_(rel)^(2)]_("initial")` `-mg2R=1/2mu(V+v)^(2)-1/2muv_(0)^(2)`..............i `mu=((mM)/(m+M))` and `v=sqrt(gR)` As there is no EXTERNAL forces acting on the system in HORIZONTAL direction. the linear momentum of the system should be conserved. i.e. `mv_(0)=m(V-v)+MV` `V=(m(v_(0)+v))/((m+M))` ............ii from eqn i and ii we get `implies v_(0)=sqrt((5+(4m)/M))gR`