Ablock of mass m is attached to a spring whose spring constantis k. the other end of the spring isfixed so that when the spring is unstretched, the mass is locate. assume that the +x direction is to the right.
the mass is now pulled to the right a distance a beyond the equilibrium position and released, at time t=0, with zero initial velocity.

assume that the vertical forces acting on the block balance eachother and that the tension of the spring is, in effect, the onlyforce affecting the motion of the block. therefore, the system willundergo simple harmonic motion. for such a system, theequation of motion is

a(t)=-\frac{k}{m}x(t),

(a) at what time t_1 does the block come back to its originalequilibrium position (x=0) for the first time?
express your answer in terms of someor all of the variables: a, k, and m.

(b) find the velocity v of the block as a function of time.
express your answer in terms of someor all of the variables: k, m, a, and t.

(c) find the acceleration a of the block as a function of time.
express your answer in terms of someof all of the variables: k, m, a, and t.

(d) specify when the magnitude of the acceleration of the blockreaches its maximum value. consider the following options:
only once during one period of motion,
when the block's speed is zero,
when the block is in the equilibrium position,
when the block's displacement equals either a or -a,
when the block's speed is at a maximum.
(e) find the kinetic energy k of the block as a function of time.
express your answer in terms of someor all of the variables: k, m, a, and t.

(f) find k_max, the maximum kinetic energy ofthe block.
express your answer in terms of someor all of the variables: k, m, and a.

(g) the kinetic energy of the block reaches its maximum when whichof the following occurs?

thedisplacement of the block is zero.
thedisplacement of the block is a.
theacceleration of the block is at a maximum.
the velocityof the block is zero.