Determine the average velocity of a projectile between the instants...

The average velocity is defined as the ratio of total displacement to total time taken.

Let the two instants when the projectile crosses half the maximum height be $t_1$ and $t_2$.

Since the vertical positions at these two instants are equal, the vertical displacement is zero.

The horizontal velocity of the projectile remains constant throughout the motion and is equal to $u \cos\theta$.

The horizontal displacement between the two instants is $\Delta x = u \cos\theta (t_2 - t_1)$.

The average velocity is $v_{avg} = \dfrac{\Delta x}{\Delta t} = \dfrac{u \cos\theta (t_2 - t_1)}{t_2 - t_1} = u \cos\theta$.

The direction of the average velocity is horizontal in the plane of projection.