Final answer:
Proving that the trajectory of a projectile is parabolic involves using kinematic equations for horizontal and vertical motion to isolate time and substituting into the vertical motion equation, resulting in a quadratic formula form y = ax + bx², where a and b are constants.
Step-by-step explanation:
Proving the Parabolic Trajectory of a Projectile
To prove that the trajectory of a projectile is parabolic, we begin with two kinematic equations that describe the motion in the horizontal (x) and vertical (y) directions. For the horizontal motion, our given equation is x = Voxt where Vox is the initial horizontal velocity and t is the time. For the vertical motion, the equation is y = Voyt - (1/2)gt2 where Voy is the initial vertical velocity and g is the acceleration due to gravity.
To find the trajectory equation, we solve the equation for x to isolate the parameter t: t = x/Vox. Substituting this expression for t into the equation for y, we get the parabolic trajectory equation: y = ax + bx2 where a = Voy/Vox and b = -g/(2Vox2).
This confirms the fact that the projectile follows a parabolic path which is a characteristic of objects under gravity when air resistance is negligible.