Final answer:
The trajectory of a projectile is proven to be parabolic by solving for time t from the horizontal motion equation and substituting into the vertical motion equation. The final equation y = ax + bx², where a and b are constants, signifies a parabolic path.
Step-by-step explanation:
The question involves proving that the trajectory of a projectile is parabolic, represented by the form y = ax + bx². To show this, let's consider the motion of the projectile having initial velocities Vox in the horizontal direction and Voy in the vertical direction, and is acted upon by gravity g. Given the equation for horizontal motion x = Vox t, we can solve for t and get t = x / Vox. Substituting this into the vertical motion equation y = Voy t - (1/2)gt² gives us:
y = Voy (x / Vox) - (1/2)g (x / Vox)²
This simplifies to:
y = (Voy / Vox) x - (1/2)g/Vox² x²
Here, the constants a and b can be defined as a = Voy / Vox and b = - (1/2)g/Vox², confirming that the equation is indeed of the form y = ax + bx² which represents a parabola.