An object is attached to the end of a vibrant spring.

Displacement of the object from its equilibrium position is .

Where is measured in seconds and is measured in centimeters.

(a)

Graph :

Graph the functions , and .

Observe the graph,

Displacement function lies between the curves and .

Prove it by theoritical approach.

The range of the  is .

.

Multiply the inequality each side by .

.

(b)

Find the maximum value of the displacement using the graph.

Observe the graph,

Maximum value of the displacement is about cm and it is occured at the time sec.

It occurs just before the displacement function touches the graph of .

(c)

Velocity of the object is the derivative of the displacement function.

.

Differentiate on each side with respect to .

.

Object attains its equilibrium position, when the displacement is zero.

.

Consider .

First time it reaches equlibrium position when .

Find the velocity at .

Velocity of the object when it reaches its equilibrium position is cm/sec.

(d)

Find the time at which the displacement is not more than cm.

Observe the graph,

The displacement is cm at time sec.

Hence the displacement of the particle is no more cm after sec.

Graph of the functions , and .

Maximum value of the displacement is about cm at the time sec.

Velocity of the object when it reaches its equilibrium position is cm/sec.

The displacement of the particle is no more cm after sec.