Final answer:
The question focuses on harmonic motion in physics, relating to the displacement and velocity of an object in sinusoidal oscillation based on time, amplitude, and other initial conditions.
Step-by-step explanation:
The question involves analyzing the motion of a continuous-time signal in a form that suggests simple harmonic motion (SHM). The amplitude X determines the maximum displacement, and the function x(t) = X cos(t) describes the displacement over time. Since the period T is the time it takes for the motion to repeat, at t = T, the displacement returns to its initial value.
The motion follows oscillatory behavior, as exemplified by the fact that at t = 0, displacement x equals the amplitude X, and at t = T, the displacement x repeats its initial value because cos(2π) = 1. To find the velocity as a function of time, differentiation is used; for instance, the given derivative of a cosine function yields a sine function representing velocity.
Problems related to wave motion or oscillatory behavior may also include considerations such as phase shifts, represented by φ in the wave function y(x, t) = A sin(kx - ωt + φ). When integrating to find displacement or velocity, initial conditions come into play, such as initial velocity v0 and constant acceleration a in the expression x(t) = ∫ (v0 + at)dt + C₂. Recognizing that C₂ = 0 when the initial displacement is zero provides the complete solution for displacement over time.