**Solution:**

To solve this problem, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of a system remains constant if no external forces are acting on it.

Given:
- Mass of the bullet (m1) = 250 g = 0.25 kg
- Mass of the bag of sand (m2) = 6.75 kg
- Initial velocity of the bullet (u1) = 200 m/s
- The bag is suspended on a ceiling by a string.

We need to find the vertical height by which the bag will swing from its original position after the impact.

**Step 1: Calculate the initial kinetic energy of the bullet.**

The initial kinetic energy of the bullet is given by the formula:

KE1 = (1/2) * m1 * u1^2

Plugging in the values, we get:

KE1 = (1/2) * 0.25 kg * (200 m/s)^2

**Step 2: Calculate the final kinetic energy of the bullet and the bag of sand.**

After the bullet is fired into the bag of sand, both the bullet and the bag acquire a common velocity (v) due to the principle of conservation of momentum.

By using the principle of conservation of momentum, we can calculate the final velocity (v) as:

m1 * u1 = (m1 + m2) * v

Plugging in the values, we get:

0.25 kg * 200 m/s = (0.25 kg + 6.75 kg) * v

**Step 3: Calculate the final kinetic energy of the bullet and the bag of sand.**

The final kinetic energy of the bullet and the bag is given by the formula:

KE2 = (1/2) * (m1 + m2) * v^2

Plugging in the values, we get:

KE2 = (1/2) * (0.25 kg + 6.75 kg) * (v)^2

**Step 4: Calculate the change in kinetic energy.**

The change in kinetic energy (ΔKE) is given by the formula:

ΔKE = KE2 - KE1

**Step 5: Calculate the change in potential energy.**

The change in potential energy (ΔPE) is equal to the change in kinetic energy (ΔKE) according to the principle of conservation of mechanical energy.

ΔPE = ΔKE

**Step 6: Calculate the vertical height.**

The change in potential energy (ΔPE) is given by the formula:

ΔPE = m2 * g * h

Where m2 is the mass of the bag, g is the acceleration due to gravity, and h is the vertical height.

Plugging in the values, we get:

ΔPE = 6.75 kg * 9.8 m/s^2 * h

Since ΔPE = ΔKE, we can equate the two equations:

6.75 kg * 9.8 m/s^2 * h = ΔKE

**Step 7: Solve for h.**

To find the vertical height, we need to solve the equation obtained in the previous step for h.

h = ΔKE / (6.75 kg * 9.8 m/s^2)

Substituting the values of Δ