We have a collision (conservation of momentum), kinetic energy, and gravitational potential energy.

Conservation of momentum:

$\overline{){{\mathbf{m}}}_{{\mathbf{1}}}{{\mathbf{v}}}_{{\mathbf{01}}}{\mathbf{+}}{{\mathbf{m}}}_{{\mathbf{2}}}{{\mathbf{v}}}_{{\mathbf{02}}}{\mathbf{=}}{{\mathbf{m}}}_{{\mathbf{1}}}{{\mathbf{v}}}_{\mathbf{f}\mathbf{1}}{\mathbf{+}}{{\mathbf{m}}}_{{\mathbf{2}}}{{\mathbf{v}}}_{\mathbf{f}\mathbf{2}}}$

Conservation of energy:

$\overline{){{\mathbf{K}}}_{{\mathbf{0}}}{\mathbf{+}}{{\mathbf{U}}}_{{\mathbf{0}}}{\mathbf{+}}{{\mathbf{W}}}_{{\mathbf{nc}}}{\mathbf{=}}{{\mathbf{K}}}_{{\mathbf{f}}}{\mathbf{+}}{{\mathbf{U}}}_{{\mathbf{f}}}}$, where W_{nc} is the work done by non-conservative forces such as friction.

**(a)**

We'll use conservation of momentum.

Let the bullet be 1 and block 2:

m_{1}v_{01} + m_{2}v_{02} = m_{1}v_{f1} + m_{2}v_{f2}

bullet and block fuse to make a single mass;

A small block of mass 2m initially rests on a track at the bottom of the circular, vertical loop-the-loop shown above, which has a radius *r*. The surface contact between the block and the loop is frictionless. A bullet of mass *m* strikes the block horizontally with initial speed v_{0} and remains embedded in the block as the block and bullet circle the loop. Determine each of the following in terms of m, v_{0}, r, and g.

(a) The speed of the block and bullet immediately after impact

(b) The kinetic energy of the block and bullet when they reach point *P* on the loop

(c) The minimum initial speed v_{min} of the bullet if the block and bullet are to successfully execute a complete circuit of the loop

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