In SHM, the distance from equilibrium where the velocity is half the maximum can be found using conservation of energy. The distance from equilibrium where the acceleration is half its maximum is simply half the amplitude of the SHM.
The question concerns the physics concept of simple harmonic motion (SHM). When a mass attached to a spring is stretched and released, it oscillates around the equilibrium position. For part (a), to find the distance from equilibrium where the velocity is half the maximum velocity, we must use the relationship between velocity and displacement in SHM. At half the maximum velocity, the kinetic energy is one-fourth of the maximum (since kinetic energy is proportional to the square of the velocity), hence the potential energy is three-fourths of the maximum. Conservation of energy can then be used to find the displacement since the total energy in SHM (sum of kinetic and potential energies) is constant.
For part (b), to find the distance where the acceleration is half its maximum, we can use Hooke's Law, which states that the force and hence the acceleration on the mass is proportional to its displacement from equilibrium. Acceleration being half the maximum implies that the displacement is also half the maximum.
Hence, if A is the amplitude, the block will have half the maximum acceleration at a displacement of A/2 from equilibrium.