Final answer:
The trajectory equation y = ax + bx² denotes a parabolic path, and the proof involves substituting the time 't' from the equation x = Vot into y = Voyt - (1/2)gt², illustrating the parabolic nature of projectile motion.
Step-by-step explanation:
The question refers to the trajectory of a projectile, which is represented by the equation y = ax + bx², where 'a' and 'b' are constants. A vertical parabola that opens downward suggests a negative 'b' value, resulting in a downward trajectory. To prove that the trajectory is parabolic, we need to solve the horizontal motion equation x = Vot for 't' and then substitute it into the vertical motion equation y = Voyt - (1/2)gt² to find an expression for 'y' in terms of 'x'
When this substitution is made and simplified, a quadratic equation in 'x' is obtained, confirming that the trajectory of a projectile is indeed parabolic. The graph for the equation y = ax + b will depend on the sign of 'b': upward sloping if 'b' is positive, horizontal if 'b' is zero, and downward sloping if 'b' is negative, as per Figure 12.4.