Motion in two and three dimensions question?

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Galileo's principle states that projectiles launched at angles equidistant from 45 degrees will have equal ranges if air resistance is negligible. The discussion involves deriving the range of a projectile using its horizontal and vertical components of velocity, defined as vx = v*cos(q) and vy_0 = v*sin(q). The time of flight is determined by setting the vertical displacement to zero, leading to the equation t = 2vy_0/g. By calculating the horizontal distance covered during this time, the relationship between the angle of projection and range can be established. Ultimately, the goal is to manipulate the range formula to demonstrate Galileo's assertion.
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Galileo shows that, if any effects due to air resistance are ignored, the ranges for projectiles on a level field whose angles of projection exceed or fall short of 45 degrees by the same amount are equal. Prove this result.

A: So ,I tried these vx = v*cos(q) //q is the shooting angle, vx is the speed in horizontal direction
vy_0 = v*sin(q) //original vertical speed
vy = vy_0 - gt // the projectile is pulled down by the gravity
----
x(t) = vxt
y(t) = vy_0t - 0.5gt^2
----
when the projectile hits the ground, y is 0
vy_0t = 0.5gt^2
vy_0 = 0.5gt
t = 2vy_0/g
How do I relate these altogether to prove the Galileo thing? Please.
 
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You have found the duration of the flight. Now find the horizontal distance covered in the flight; this is the range in the question. The range will depend on the initial velocity and the angle of projection, so manipulate its formula to show what is required.
 
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