Amass suspended from a spring is oscillating up and down as indicated. consider the following possibilities. a at some point during the oscillation the mass has zero velocity but its acceleration is non-zero (can be either positive or negative); b at some point during the oscillation the mass has zero velocity and zero acceleration; c at some point during the oscillation the mass has non-zero velocity (can be either positive or negative) but has zero acceleration; d at all points during the oscillation the mass has non-zero velocity and has nonzero acceleration (either can be positive or negative). which possibility or possibilities occur?

In a bouncing spring, the mass will have zero velocity and non-zero acceleration at the top and bottom end of motion.

It is defined as the rate of change in velocity or change speed/direction of the object.

An object can have zero velocity but non-zero acceleration. It can be understood by 2 examples,

Therefore, in a bouncing spring, the mass will have zero velocity and non-zero acceleration at the top and bottom end of motion.

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A mass suspended from a spring is oscillating up and down, (as stated but not indicated).

A). At some point during the oscillation the mass has zero velocity but its acceleration is non-zero (can be either positive or negative).  Yes.  This statement is true at the top and bottom ends of the motion.

B). At some point during the oscillation the mass has zero velocity and zero acceleration.  No.  If the mass is bouncing, this is never true.  It only happens if the mass is hanging motionless on the spring.

C). At some point during the oscillation the mass has non-zero velocity (can be either positive or negative) but has zero acceleration.  Yes.  This is true as the bouncing mass passes through the "zero point" ... the point where the upward force of the stretched spring is equal to the weight of the mass.  At that instant, the vertical forces on the mass are balanced, and the net vertical force is zero ... so there's no acceleration at that instant, because (as Newton informed us), A = F/m .  

D). At all points during the oscillation the mass has non-zero velocity and has nonzero acceleration (either can be positive or negative).  No.  This can only happen if the mass is hanging lifeless from the spring.  If it's bouncing, then It has zero velocity at the top and bottom extremes ... where acceleration is maximum ... and maximum velocity at the center of the swing ... where acceleration is zero.