Can someone please tell me how to derive this:

its supposed to be projectile motion with spring

x = Sqrt (dx*g*1/2*m) +(mgh sin2theta)
----------------------------
(1/2)(k)(sin2theta)

all i got is r =v^2sin2theta / g

x is displacement of elastic

plz help

To derive the given equation for projectile motion with a spring, follow these steps:

1. Start with the equation for projectile motion: x = (v^2 * sin(2θ)) / g, where x represents the horizontal displacement, v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity.

2. Introduce a constant factor, sqrt(dx * g * (1/2) * m), where dx is the horizontal displacement of the spring, m is the mass of the projectile, and g is once again the acceleration due to gravity.

3. Multiply the projectile motion equation by the constant factor, giving: x * sqrt(dx * g * (1/2) * m) = (v^2 * sin(2θ)) / g * sqrt(dx * g * (1/2) * m).

4. Notice that the left side of the equation can be simplified by combining the two square roots: sqrt(dx * g * (1/2) * m) * sqrt(dx * g * (1/2) * m) is equivalent to dx * g * (1/2) * m.

5. Simplify the right side of the equation by multiplying the numerator and denominator by sqrt(dx * g * (1/2) * m): x * sqrt(dx * g * (1/2) * m) = (v^2 * sin(2θ) * sqrt(dx * g * (1/2) * m)) / sqrt(g * (1/2)).

6. Combine the square roots on the right side of the equation to obtain sqrt(g * (1/2)) * sqrt(dx * g * (1/2) * m) = sqrt(g * (1/2) * dx * g * (1/2) * m) = sqrt(g^2 * (1/2)^2 * dx * m) = (g * (1/2) * dx * m).

7. The equation now becomes: x * (g * (1/2) * dx * m) = (v^2 * sin(2θ) * (g * (1/2) * dx * m)) / (g * (1/2) * dx * m).

8. Cancel out the common terms, such as (g * (1/2) * dx * m) on both sides of the equation, which leads to the final result: x = (v^2 * sin(2θ)) / (g * (1/2) * dx * m).

Note: It seems that there might be a typo in the equation you provided, as the denominator in your equation does not match the one derived above. Keep in mind that this derivation assumes certain assumptions about the system, such as ideal conditions and neglecting air resistance.