Final answer:
To prove a projectile's trajectory is parabolic, start with the motion equations x = V0xt and y = V0yt - (1/2)gt^2, solve for t from the first equation, and substitute it into the second to eliminate the time variable. This yields the parabolic form y = ax + bx^2.
Step-by-step explanation:
Proving the Parabolic Trajectory of a Projectile
To prove that the trajectory of a projectile is parabolic, we start with the definition of the parabola as the path followed by an object under the influence of constant acceleration, such as gravity, when air resistance is negligible. The equations x = V0xt for horizontal motion and y = V0yt - (1/2)gt2 for vertical motion describe the projectile's path. To find the equation in the form of a parabola, which is y = ax + bx2, we need to eliminate the time variable, t.
From the equation for horizontal motion, we can express t as x/V0x. Substituting this into the vertical motion equation yields:
y = V0y(x/V0x) - (1/2)g(x/V0x)2
Rearranging terms gives us an equation of the parabolic form:
y = (V0y/V0x)x - (1/2)g/V0x2x2
This equation clearly has the structure y = ax + bx2, where a and b represent constants that depend on the projectile's initial velocity components and the acceleration due to gravity.