Final answer:
None of the provided options accurately describe an object's velocity as a function of time considering both gravity and a resistive force. This scenario requires solving a differential equation that accounts for both forces.
Step-by-step explanation:
When considering an object moving vertically with an initial velocity vₒ under the influence of gravity and a resistive force proportional to its velocity, the correct formula for its velocity as a function of time is not simply v(t) = vₒ - gt. This equation only accounts for gravity and not the resistive force. The resistive force, which is a function of the velocity (f = -bv), works to decelerate the object in addition to gravity. Therefore, the velocity equation needs to also include a term that accounts for this resistance.
An accurate model that includes both gravity and a resistive force would require solving a differential equation where a(t) = dv/dt = -g - bv, which is not straightforward and does not match any of the given options a) through d). All given options are missing the proper treatment of the resistive force term, which would involve more complex mathematics, potentially including an exponential function of time, but not in the simple form of the options provided.
Thus, none of the provided options a), b), c), or d) accurately describe the velocity of the object as a function of time for the scenario where both gravity and a resistive force proportional to the velocity are present.