Conservation of Momentum Problem - Mechanical and Kinetic Energies

[Ascendant0]

Homework Statement

A large block of wood of mass M = 5.4 kg is hanging from two long cords. A bullet of mass m = 9.5 g is fired into the block, coming quickly to rest. The block & bullet then swing upward, their center of mass rising a vertical distance h = 6.3 cm before the pendulum comes momentarily to rest at the end of its arc. What is the speed of the bullet just prior to the collision?

Relevant Equations

$1/2(m + M)V^2 = (m + M)gh$

I'm confused on this problem, as I feel they state two completely contradictory things in the explanation of how to solve it. The first statement that I feel contradicts the second is this:

"We can see that the bullet’s speed v must determine the rise height h. However, we cannot use the conservation of mechanical energy to relate these two quantities because surely energy is transferred from mechanical energy to other forms (such as thermal energy and energy to break apart the wood) as the bullet penetrates the block."

That explanation makes complete sense to me. But then, I continue reading, and two paragraphs later, the text then states:

"Then conservation of mechanical energy means that the system’s kinetic energy at the start of the swing must equal its gravitational potential energy at the highest point of the swing."

And right after that paragraph, they go on to relate the two quantities with kinetic and potential energy equations (the ones in the "relevant equations" posted above). So, the first paragraph states we can't use the conservation of mechanical energy, then the second one here tells you TO use the conservation of mechanical energy, and applies it to the problem. This is copied verbatim from the text, so hopefully you can see how this is confusing me.

I would think it would be the kinetic energy of the bullet that is lost to thermal and other energies, and so the potential energy would be less than it, and that seems to be what the first paragraph indicates. But then, the second one completely throws me off.

Can someone please make sense of this and how this is not contradictory? Also, if you could please clarify what equations cannot be used in situations like this so I know for future reference what they were actually referring to, and which ones are ok to use?

 

[haruspex]

You need to consider two distinct phases.
In the first phase, the bullet becomes lodged in the block. Yes, a lot of mechanical energy is lost to heat, but it is still possible to determine the speed with which the bullet+block start to move together.
In the second phase, the moving block + bullet swing up to some height. In this phase, mechanical energy is largely conserved.
Strictly speaking, the two phases overlap slightly, but you can ignore that.

 

[Ascendant0]

You need to consider two distinct phases.
In the first phase, the bullet becomes lodged in the block. Yes, a lot of mechanical energy is lost to heat, but it is still possible to determine the speed with which the bullet+block start to move together.
In the second phase, the moving block + bullet swing up to some height. In this phase, mechanical energy is largely conserved.
Strictly speaking, the two phases overlap slightly, but you can ignore that.

Thank you. I do see what you're saying. It's making more sense to me now.