Assume the collision between the bullet and block is a perfe

Assume the collision between the bullet and block is a perfect inelastic collision. Obtain an expression for the velocity of the block-bullet system after the bullet is lodgcd in the block, but before the pendulum swings upwards. Express in terms ofM,m, and v_0. What is the change in vertical position, call it h, for the bullet-block system as they reach theta_max, in terms of L and theta_max ? Use your answers to a), b) and another fundamental principle of physics to obtair an expression for the initial velocity of the bullet in terms of L, g, M, m and theta_max.

Solution

Since the collision is inelastic, KE of the system will not be conserved. Of course the total energy of system is conserved.

a) Conserving momentum we have

mv_o= (M+m)v => v= mv_o/(M+m)

b) h= L(1-cos_max)

c)Once the bullet sticks to the mass M, the system will behave as a pendulum with mass M+m which is given a velocity v(as determined in part 1).

Applying conservation of energy to this new system,

Initial KE + initial PE = final KE +final PE

0.5x(M+m)(mv_o/(M+m))² + 0 = 0 + (M+m)g(L-Lcos_max)

=> v_o= (2gL(1-cos_max))^0.5(M+m)/m -----(1)