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I am changing symbols for speed and angle.

Let us say the the projectile
is fired into air at an angle of ** Ф** with the horizontal and with an initial
speed of

There is a deceleration in the vertical direction = - g

Speed of projectile in the vertical direction is : V = u + a * t

=> Vy = u sin Ф - g t --
equation 1

Speed in the horizontal direction is a constant as there is no acceleration:

=> Vx = u cos Ф + 0 * t -- equation 2

There is no acceleration in X-direction.

Displacement = Distance traveled is : s = u t + 1/2 a t²

=>
Sx = x = (u cos Ф) t
--- equation 3

t = x / (u cos Ф)

Sy = y
= (u Sin Ф) t - 1/2 g t²

y =
(u sin Ф) (x / u cos Ф) - 1/2 g (x² / u² Cos² Ф)

**
y = x tan Ф -
g x² / (2u²Cos²Ф) ** ---
equation 4

This is the equation of motion of the projectile, in 2 dimensions. This is the locus.

====================================

The maximum height reached by projectile is H.

y = H when Vy becomes 0. =>
Vy = 0 = u sin Ф - g t

=> ** t = u sin Ф /
g** = time interval to reach the maximum height.

H = u Sin Ф * (u sin Ф / g)
- 1/2 g (u SinФ/g)²

Hence, **H = u² Sin² Ф / 2 g** -- equation
5

y = 0 => x = R = Range of the projectile , where it lands on ground.

Substituting in equation 4,
we get

**R = u² Sin 2Ф / g
-- equation 6**