Find the average velocity of a projectile between the instants it first crosses half the maximum height and then again does the same. It is projected with a speed u at an angle θ with the horizontal. (no need to calculate, just work it out mentally!).
If we focus on the part of motion after the projectile has crossed half the projectile has crossed half the maximum height and has velocity v and makes angled with the x-axis.
Average velocity = total displacementtotal time
Total displacement is happening on the x-axis only
Which if I consider this part of motion as a new projectile will be its range I know range is nothing but Horizontal component of velocity × Time period
R = v cos α × T
Time period in this case will be 2 v sin αg
So R = v cos α × 2v sinαg
Now vavg = RT = v cosα × 2 v sinαg2v sinαg = v cosα = u cosθ.