Final answer:
When the displacement of an object in simple harmonic motion is zero, it reaches its maximum speed; hence, if the speed is v at half amplitude, the speed at zero displacement is 2v (option C).
Step-by-step explanation:
In the context of an object undergoing simple harmonic motion described by D(t) = A sin(wt), where A is the amplitude and w is the angular frequency, and the speed of the object is v when its displacement is half of the amplitude, the speed of the object when its displacement is zero relates to the maximum speed the object can have during its motion.
Simple harmonic motion can be visualized as the projection of uniform circular motion. When an object is undergoing simple harmonic motion and its displacement is zero, it is at the equilibrium position, which coincides with the object having its maximum speed. Therefore, the object's speed at a displacement of zero is the maximum speed Umax, which is equal to Aw.
As given in the provided reference materials, when the displacement is half the amplitude, the speed is half the maximum speed, i.e., v = ±Umax/2. Therefore, if we let v be the speed when the displacement is A/2, then at zero displacement, the speed must be Umax, which is 2v. Thus, the speed of the object in terms of v when its displacement is zero is 2v, which corresponds to option C).