Final answer:
The trajectory of a projectile is parabolic, derived by solving the horizontal motion equation for time and substituting into the vertical motion equation, resulting in a parabolic equation y = ax + bx².
Step-by-step explanation:
The question you've asked pertains to the trajectory of a projectile and involves deriving the equation of its path. To demonstrate that the trajectory is parabolic with a vertical transverse axis, we manipulate the expressions for horizontal and vertical motion.
The horizontal distance (x) covered by the projectile is given by x = V0xt, where V0x is the horizontal component of the initial velocity, and t is the time elapsed. For the vertical position (y), the expression is y = Voyt - (1/2)gt², where Voy is the vertical component of the initial velocity, and g is the acceleration due to gravity.
To find the trajectory's equation, we solve for t from the horizontal motion equation and substitute it into the vertical motion equation. With this substitution, we get an equation of the form y = ax + bx², where a and b are constants that depend on the initial velocity components and gravity. This resulting equation represents a parabola, confirming the parabolic trajectory of the projectile.