Prove ThatThe velocity of the projectile at any t during its motion is given byv = Square Root (u^2+g^2*t^2-2ugt*Sin Theta)

to prove this qusetion requires simple vector addition.

just imagine that at any instant the net velocity of the projectile is the superposition of two velocities:

1. the projection velocity u inclined at θ to the horizontal.

2. a vertically downward veloctiy equal to gt which a freely falling body would acquire when it has fallen for time t.

the angle between these vectors equals 90° + θ.

so the net velocity at any time comes out to be v = {u² + g²t² - 2ugtsinθ}^(1/2).