Final answer:
Wave functions valid as solutions to Schrödinger's equation must be continuous, normalizable, and satisfy the equation's requirements. Acceleration, such as gravity, is a vector quantity, while charge interactions indicate the charge on Z could be positive or neutral.
Step-by-step explanation:
The validity of a wave function as a solution to Schrödinger's equation depends on certain properties. It must be continuous, normalizable, and it must satisfy the time-independent Schrödinger equation for the given potential. The provided options are not detailed enough to determine if any are valid wave functions without additional context or notation. In Physics, when discussing the Schrödinger equation, potential options for valid wave-function solutions would typically look more like mathematical functions, for instance, ψ(x) = A e(-x2)/a2, where A is a normalization constant.
For the question about vector quantities, the acceleration of gravity is a vector because it has both a magnitude and a direction. The other items listed, such as the age of the Earth or the boiling point of water, are scalar quantities because they have magnitude only, with no direction.
Concerning the question about charge interactions, if balloon Y repels negatively charged balloon X, we can infer that balloon Y is also negatively charged. Therefore, object Z, which is attracted to balloon Y, could either be positively charged or have no net charge (neutral). Thus, the charge on Z could be positive or neutral.