Simple Conservation of Energy Question 

Deduce that the speed $V$ with which the bullet strikes the plasticine is about $160 \ m/s$

Using the conservation of energy, I equated the kinetic energy of the bullet to the potential energy of both the plasticine and bullet, at maximum height when their kinetic energy is equal to zero:
$$mgh = \frac{1}{2}mv^2$$
$$(0.38 + (5.2 \cdot 10^{-3}))(g)(0.24) = (0.5)(5.2 \cdot 10^{-3})(V^2)$$
$$\therefore V = 18.68 \ m/s$$
What is wrong with my conceptual understanding?
 A: In your solution, two things are wrong: 


*

*The final energy of the system consists of the potential energy of the bob and the bullet. If the mass of the bob is $M$ and that of the bullet is $m$ then the final potential energy is $(M+m)gh$ and not $mgh$ or $Mgh$. Though if the mass of the bullet is very tiny then you can approximate it to $Mgh$ but not to $mgh$.

*The collision between the bullet and the bob is completely inelastic. Since they both move together after the collision you can clearly say that the coefficient of the restitution is zero and the collision is completely inelastic. In such a collision the initial total kinetic energy doesn't equate the final total kinetic energy. Some of the initial kinetic energy goes into the potential energy of the structure. (You can appreciate that the structure of the bob changes due to the collision - this change consumes some energy.) So you can't equate the initial total kinetic energy with the final total gravitational potential energy because some of the initial kinetic energy goes into the structural potential energy of the bob. 
The right way to resolve the issue can be to use momentum conservation during the collision in the horizontal direction. Because no force acts horizontally on the system at the time of the collision. And then conservation of mechanical energy is applicable. 
A: When the bullet hits the plasticine and embedded inside the plasticine, there is friction exerts on the bullet and hence heat is dissipated to the surrounding.
Therefore, conservation of mechanical energy cannot be applied when the bullet is embedding the plasticine. 
But as for the bullet and the plasticine together displaced upward, conservation of mechanical energy is still applicable since there is no dissipative force applied during this process.
So, the problem is how to relate the situation before the bullet hits the plasticine and after it is embedded inside the plasticine. During this process, there is no external force acting on the system. Therefore, conservation of linear of momentum can be applied in this process.
