**Problem Analysis:**

In this problem, we are given that a bullet of mass 90g is moving with a velocity of 300 m/s and it penetrates into a block of wood to a depth of 10 cm. We need to find the emerging velocity of the bullet when it is fired into a similar type of wood block with a thickness of 5 cm.

**Solution:**

Let's solve this problem step by step:

1. **Initial Kinetic Energy of the Bullet:**

We know that the kinetic energy (KE) of an object is given by the equation:

KE = (1/2)mv^2

Where m is the mass of the object and v is its velocity.

In this case, the mass of the bullet (m) is 90g, which is equal to 0.09 kg, and its velocity (v) is 300 m/s.

So, the initial kinetic energy of the bullet is:

KE = (1/2)(0.09)(300)^2 = 1215 J

2. **Work Done by the Bullet:**

When the bullet penetrates into the wood block, it does work against the resistance offered by the wood. This work is equal to the change in kinetic energy of the bullet.

Since the bullet comes to rest after penetrating to a depth of 10 cm, its final velocity (vf) is 0 m/s.

So, the work done by the bullet is:

Work = KE - 0 = 1215 J

3. **Work-Energy Principle:**

According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy.

Work = Change in Kinetic Energy

So, the work done by the bullet is equal to the change in its kinetic energy:

Work = 1215 J

4. **Work Done by the Bullet on the Second Wood Block:**

When the bullet is fired into the second wood block, it again does work against the resistance offered by the wood. This work is equal to the change in kinetic energy of the bullet.

Since the bullet emerges with a velocity (ve) after penetrating to a depth of 5 cm, we need to find the value of ve.

So, the work done by the bullet on the second wood block is:

Work = (1/2)(0.09)(ve)^2 - 0 = 1215 J

5. **Final Kinetic Energy of the Bullet:**

The final kinetic energy of the bullet can be calculated using the work done on it:

(1/2)(0.09)(ve)^2 = 1215 J

Simplifying the equation, we get:

(ve)^2 = (1215 J) * (2/0.09)

(ve)^2 = 27,000

Taking the square root of both sides, we get:

ve = √(27,000) = 164.32 m/s

6. **Conclusion:**

Therefore, the emerging velocity of the bullet when fired into the second wood block is approximately 164.32 m/s.