**Given information:**
- Initial velocity of the particle, u = 10 m/s
- Time taken to reach the ground, t = 5 s

**To find:**
(i) Height of the tower
(ii) Striking velocity of the particle at the ground
(iii) Distance traversed by the particle
(iv) Average speed and average velocity of the particle

**Solution:**

**i) Height of the tower:**
When the particle is projected vertically up, it will reach the highest point and then fall back to the ground. The total time of flight will be twice the time taken to reach the highest point.

Let's assume the height of the tower is h.

Using the equation of motion: s = ut + (1/2)at², where s is the distance covered, u is the initial velocity, t is the time taken, and a is the acceleration.

When the particle is projected upwards:
- Initial velocity, u = 10 m/s
- Final velocity, v = 0 m/s (at the highest point)
- Time, t₁ = t/2 (half the total time of flight)

Using the equation of motion, s = ut + (1/2)at², we have:
0 = 10(t/2) - 5(t/2)²
0 = 5t - 5(t²/4)
0 = 5t - (5/4)t²
0 = 20t - 5t²

Simplifying the equation, we get:
t(20 - 5t) = 0
t = 0 or t = 4

Since time cannot be zero, the particle will take 4 seconds to reach the highest point.

Therefore, the total time of flight, t_total = 2t = 2(4) = 8 s

Using the equation of motion, s = ut + (1/2)at², we have:
h = 10(4) - 5(4²/2)
h = 40 - 40
h = 0

Therefore, the height of the tower is 0 m.

**ii) Striking velocity of the particle at the ground:**
The striking velocity of the particle at the ground is the final velocity when it reaches the ground.

Using the equation of motion, v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.

When the particle is falling down:
- Initial velocity, u = 0 m/s (at the highest point)
- Acceleration, a = 9.8 m/s² (acceleration due to gravity)
- Time, t = t_total = 8 s (total time of flight)

Using the equation of motion, v = u + at, we have:
v = 0 + 9.8(8)
v = 78.4 m/s

Therefore, the striking velocity of the particle at the ground is 78.4 m/s.

**iii) Distance traversed by the particle:**
The distance traversed by the particle is the total vertical distance covered during the entire motion.

Since the height of the tower is 0 m and the particle falls back to the ground, the distance traversed by the particle is equal to the height of the tower.

Therefore, the distance