**Problem Analysis:**

In this problem, a bullet is fired into a bag of sand that is suspended from the ceiling by a string. The bullet has a mass of 250g and a speed of 200m/s. The bag of sand has a mass of 6.75kg. We are asked to find the vertical height to which the bag will swing after the impact.

To solve this problem, we can use the principle of conservation of mechanical energy. The initial mechanical energy of the system is equal to the final mechanical energy of the system. The initial mechanical energy is the sum of the kinetic energy of the bullet and the potential energy of the bag of sand. The final mechanical energy is the sum of the kinetic energy and potential energy of the swinging bag of sand.

**Solution:**

**Step 1: Find the initial mechanical energy:**

The initial mechanical energy is the sum of the kinetic energy of the bullet and the potential energy of the bag of sand.

Kinetic energy of the bullet:
KE_bullet = 1/2 * mass_bullet * velocity^2

Given:
mass_bullet = 250g = 0.25kg
velocity = 200m/s

Substituting these values into the equation, we get:
KE_bullet = 1/2 * 0.25kg * (200m/s)^2

**Step 2: Find the final mechanical energy:**

The final mechanical energy is the sum of the kinetic energy and potential energy of the swinging bag of sand.

Kinetic energy of the bag of sand:
KE_bag = 1/2 * mass_bag * velocity^2

Given:
mass_bag = 6.75kg
velocity = 0m/s (at the highest point of swing, the bag momentarily comes to rest)

Substituting these values into the equation, we get:
KE_bag = 1/2 * 6.75kg * (0m/s)^2

Potential energy of the bag of sand:
PE_bag = mass_bag * g * height

Given:
mass_bag = 6.75kg
g = 9.8m/s^2 (acceleration due to gravity)
height = ?

Substituting these values into the equation, we get:
PE_bag = 6.75kg * 9.8m/s^2 * height

**Step 3: Equate the initial and final mechanical energies and solve for height:**

Setting the initial mechanical energy equal to the final mechanical energy, we have:
KE_bullet = KE_bag + PE_bag

Substituting the expressions for KE_bullet, KE_bag, and PE_bag, we get:
1/2 * 0.25kg * (200m/s)^2 = 1/2 * 6.75kg * (0m/s)^2 + 6.75kg * 9.8m/s^2 * height

Simplifying the equation, we get:
0.25kg * (200m/s)^2 = 6.75kg * 9.8m/s^2 * height

Solving for height, we find:
height = (0.25kg * (200m/s)^2) / (6.75kg * 9.8m/s^2)

Evaluating the expression, we get:
height ≈ 7.35m

Therefore, the vertical height to which the bag will swing from its original