Analysis:
When a particle is projected at an angle to the horizontal, its velocity can be resolved into two components: one along the horizontal axis and the other along the vertical axis. The vertical component of the velocity is affected by gravity, while the horizontal component remains constant throughout the motion.

Initial Velocity Components:
Given that the particle is projected at an angle of 30 degrees with the horizontal and has an initial speed of 20 m/s, we can calculate the initial velocity components as follows:

Horizontal Component (Vx): Vx = V * cosθ
Vertical Component (Vy): Vy = V * sinθ

Where V is the initial speed (20 m/s) and θ is the angle of projection (30 degrees).

Vx = 20 * cos(30) = 20 * 0.866 = 17.32 m/s
Vy = 20 * sin(30) = 20 * 0.5 = 10 m/s

Time of Flight:
The time of flight of the projectile can be determined by considering the vertical motion. The particle will reach its maximum height when the vertical component of velocity becomes zero. At this point, the velocity vector will be perpendicular to the initial velocity.

The time taken to reach the maximum height can be calculated using the equation:
Vy = u + at

Where Vy is the final vertical component of velocity, u is the initial vertical component of velocity, a is the acceleration due to gravity (-9.8 m/s²), and t is the time.

0 = 10 + (-9.8) * t
10 = 9.8t
t = 10 / 9.8 = 1.02 seconds

Since the time of flight is the total time taken for the particle to reach the maximum height and return to the ground, it will be twice the time calculated above.

Time of Flight (T) = 2 * t = 2 * 1.02 = 2.04 seconds

Therefore, the time after which the velocity vector of the projectile is perpendicular to the initial velocity is 2.04 seconds.