A projectile fired at 30$$^\circ$$ to the ground is observed to be at same height at time 3 s and 5 s after projection, during its flight. The speed of projection of the projectile is _______ m s$$^{-1}$$.
(Given $$g = 10$$ m s$$^{-2}$$)

For a projectile, if it is at the same height at two different times $$t_1$$ and $$t_2$$, then by symmetry of the trajectory, these times are symmetric about the time of maximum height.

Time of maximum height = $$\frac{t_1 + t_2}{2} = \frac{3 + 5}{2} = 4$$ s

The total time of flight is:

$$T = 2 \times 4 = 8$$ s

Using the time of flight formula:

$$T = \frac{2u\sin\theta}{g}$$

$$8 = \frac{2u\sin 30°}{10} = \frac{2u \times 0.5}{10} = \frac{u}{10}$$

$$u = 80 \text{ m/s}$$

The speed of projection is $$80$$ m s$$^{-1}$$.