**Energy Conservation Principle**

According to the law of conservation of energy, the total energy of an isolated system remains constant. In this case, we can apply the principle of energy conservation to determine the emerging velocity of the bullet.

**Initial Energy of the Bullet**

The initial energy of the bullet can be calculated using the formula:

\[E_{\text{initial}} = \frac{1}{2} m v_{\text{initial}}^2\]

Where:
- \(m\) is the mass of the bullet (90g or 0.09kg)
- \(v_{\text{initial}}\) is the initial velocity of the bullet (300m/s)

Substituting the given values, we get:

\[E_{\text{initial}} = \frac{1}{2} \times 0.09 \times 300^2\]
\[E_{\text{initial}} = 4050 \, \text{J}\]

**Energy Used to Penetrate the Wood**

The bullet penetrates into the wood, and this process requires energy. The energy used to penetrate the wood can be calculated using the formula:

\[E_{\text{wood}} = \text{Force} \times \text{Distance}\]

The force can be calculated using Newton's second law:

\[F = \text{mass} \times \text{acceleration}\]

The acceleration can be calculated using the formula:

\[a = \frac{v_{\text{final}} - v_{\text{initial}}}{t}\]

Where:
- \(v_{\text{final}}\) is the final velocity of the bullet after penetrating the wood
- \(t\) is the time taken to penetrate the wood (assumed to be the same for both cases)

Substituting the given values into the equation, we get:

\[a = \frac{212 - 300}{t}\]

Assuming the time taken to penetrate the wood is the same for both cases, we can equate the force for both cases:

\[F_1 \times 10 = F_2 \times 5\]

Simplifying the equation, we get:

\[F_1 = 2F_2\]

Since the forces are inversely proportional to the accelerations:

\[a_1 = \frac{1}{2} a_2\]

Substituting this into the acceleration formula, we get:

\[\frac{212 - 300}{t_1} = \frac{1}{2} \times \frac{212 - v_{\text{final}}}{t_2}\]

Simplifying the equation, we get:

\[\frac{212 - 300}{t_1} = \frac{106 - v_{\text{final}}}{t_2}\]

Since the time taken to penetrate the wood is the same for both cases, we can equate \(t_1\) and \(t_2\):

\[t_1 = t_2\]

Substituting this into the equation, we get:

\[\frac{212 - 300}{t_1} = \frac{106 - v_{\text{final}}}{t_2}\]

Simplifying the equation, we get:

\[\frac{212 - 300}{t_1} = \frac{106 - v_{\text{final}}