Given Data:
Initial velocity (u) = 16 m/s (upwards)
Final velocity (v) = 8 m/s (downwards)
Acceleration due to gravity (g) = 10 m/s² (downwards)

Approach:
To find the maximum height attained by the particle, we can use the equation of motion:

v² = u² + 2as

where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.

Since the particle is projected vertically upwards, the acceleration due to gravity acts in the opposite direction (downwards). So, the acceleration (a) will be -g.

Solution:
1. Calculate the time taken to reach the maximum height:
Using the equation of motion:

v = u + at

Substituting the given values, we have:
8 = 16 - 10t

Simplifying the equation:
10t = 8
t = 0.8 seconds

2. Calculate the maximum height (H) attained by the particle:
Using the equation of motion:

v² = u² + 2as

Substituting the given values, we have:
0 = 16² + 2(-10)H

Simplifying the equation:
256 - 20H = 0
H = 256/20
H = 12.8 meters

However, this is not the correct answer. The work done by air resistance is the same during upward and downward motion. This implies that the particle loses the same amount of kinetic energy during both phases.

3. Calculate the kinetic energy at the highest point:
At the highest point, the particle momentarily comes to rest and all its initial kinetic energy is converted into potential energy.

Kinetic energy at the highest point (K.E.) = 1/2 mv²
Potential energy at the highest point (P.E.) = mgh

Since the particle loses the same amount of kinetic energy during upward and downward motion, we can equate the kinetic energy at the highest point to the kinetic energy just before reaching the point of projection.

1/2 m(8)² = 1/2 m(16)²
64m = 256m
m cancels out

Therefore, the kinetic energy at the highest point is the same as the kinetic energy just before reaching the point of projection.

4. Calculate the maximum height using the equation of potential energy:
Potential energy at the highest point (P.E.) = mgh

1/2 m(8)² = mgh

Simplifying the equation:
32 = 10h
h = 32/10
h = 3.2 meters

However, this is still not the correct answer. We need to consider the work done by air resistance.

5. Consider the work done by air resistance:
The work done by air resistance is equal to the difference in potential energy between the highest point and the point of projection.

Work done by air resistance = P.E.(highest point) - P.E.(point of projection)
Work done by air resistance = mgh - 0
Work done by air resistance = mgh

Since the work done by air resistance is the same during upward and downward motion, it is equal to the potential energy at the highest point.

mgh = 32

Simplifying the equation: