Final answer:
The trajectory of a projectile is proven to be parabolic by substituting the solved time from the horizontal motion equation into the vertical motion equation, resulting in a parabolic equation y = ax + bx² where a and b are constants based on the initial velocity components and acceleration due to gravity.
Step-by-step explanation:
To prove that the trajectory of a projectile is parabolic with the form y = ax + bx², we can start by using the equations that represent the horizontal and vertical positions of a projectile. The horizontal position is given by x = Vox × t, and the vertical position is given by y = Voy × t - (1/2)gt², where Vox and Voy are the horizontal and vertical components of the initial velocity, respectively, g is the acceleration due to gravity, and t is time.
Solving for time t in the horizontal motion equation yields t = x/Vox. Substituting this value into the equation for vertical position, we eliminate the parameter t and obtain an equation that depends solely on x and y:
y = Voy × (x/Vox) - (1/2)g × (x/Vox)²
This simplifies to the parabolic form:
y = (Voy/Vox) × x - (g/2Vox²) × x²
Which can be written as y = ax + bx² where a = Voy/Vox and b = -g/2Vox² are constants.
This shows that the path of a projectile is indeed a parabola when plotted as a function of vertical displacement y against horizontal displacement x.