Final answer:
To prove the parabolic trajectory of a projectile, solve the horizontal motion equation for time and substitute it into the vertical motion equation, resulting in a parabolic form y=ax+bx². Use the quadratic formula to solve for specific time points when a = 4.90, b = -14.3, c = -20.0.
Step-by-step explanation:
To prove that a projectile's trajectory is parabolic, we start with the kinematic equations for projectile motion. Given the horizontal motion equation x = Voxt, where Vox is the horizontal component of the initial velocity, we can solve for t (time) as t = x/Vox. Then we use the vertical motion equation y = Voyt - (1/2)gt² where Voy is the vertical component of the initial velocity and g is the acceleration due to gravity. Substituting t into this equation gives us:
y = Voy(x/Vox) - (1/2)g(x/Vox)²
After simplifying, we obtain an equation of the form y = ax + bx², where a = Voy/Vox and b = -g/(2Vox²). These coefficients represent constants based on the initial velocity components and gravity, confirming that the trajectory is parabolic.
For the quadratic equation at² + bt + c = 0, we can also use the quadratic formula for solving for t. When a = 4.90, b = -14.3, and c = -20.0, the solutions to the equation give us the times at which a projectile reaches a certain position.