#### To determine

**(a)**

**To find: **The velocity and acceleration functions

#### Answer

Velocity =

c2tb2+c2t2

Acceleration =

c2b2b2+c2t232

#### Explanation

**1) Concept: **

By differentiating position function we can obtain velocity function and further by differentiating velocity function we can obtain acceleration function

**2) Formula:**

(i) velocity=v=dxdt

(ii) acceleration=a=dvdt

(iii) Chain rule: ddx(fgx=f'gx*g'(x)

(iv) Sum rule: ddxfx+gx=ddxfx+ddx(gx)

(v) Constant multiple rule: ddxCfx=Cddxfx

(vi) Constant function rule: ddxc=0

(vii) Power rule: ddxxn=nxn-1

(viii) Product rule: ddxfx*gx=fxddxgx+gxddx(fx)

**3) Given:**

Position function is x= b2+c2t2, t≥0

**4) Calculation:**

Consider the function,

x= b2+c2t2

By using formula,

v=dxdt=ddt(b2+c2t2)

By using power and chain rule,

v=12b2+c2t2ddtb2+c2t2

By using sum rule,

v=12b2+c2t2ddtb2+ddt(c2t2)

By using constant function rule and constant multiple rule,

v=12b2+c2t20+c2ddtt2

By using power rule,

v=12b2+c2t2(2c2t)

Therefore,

v=c2tb2+c2t2

Now by using formula,

a=dvdt=ddtc2tb2+c2t2=ddtc2tb2+c2t2-12

By using product rule,

a=c2tddtb2+c2t2-12+ b2+c2t2-12)ddt(c2t)

By using power rule and chain rule,

a=c2t-12b2+c2t2-32*ddt(b2+c2t2)+ b2+c2t2-12ddt(c2t)

By using sum rule and constant multiple rule,

a=c2t-12b2+c2t2-32ddtb2+ddt(c2t2)+ b2+c2t2-12(c2)ddt(t)

By using constant function rule and constant multiple rule,

a=c2t-12b2+c2t2-320+c2ddtt2+ b2+c2t2-12(c2)

By using power rule,

a=c2t-12b2+c2t2-32(2c2t)+ c2b2+c2t2-12

Taking b2+c2t2-32 common,

a=b2+c2t2-32[c2t-122c2t+ c2b2+c2t2)

a=b2+c2t2-32[-(c4t2)+ c2b2+c4t2]

a=b2+c2t2-32( c2b2)

⇒a=c2b2b2+c2t232

**Final statement:**

Velocity =

c2tb2+c2t2

Acceleration =

c2b2b2+c2t232

#### To determine

**(b)**

**To show:** Particle always moves in the positive direction

#### Answer

Particle always moves in the positive direction

#### Explanation

**1) Concept: **

By showing velocity is always positive we can show particle always moves in the positive direction

**2) Given:**

t≥0

**3) Calculation**:

To show the particle always moves in the positive direction we have to show that the velocity is always positive

From part (a), Velocity = c2tb2+c2t2

Since the denominator is always ≥ 0 only numerator can affect the sign of velocity

But we are given t≥0 and c2 being a square is positive therefore numerator is always positive,

Therefore, Velocity = c2tb2+c2t2≥0

Therefore, particle always moves in the positive direction

**Final statement:**

Particle always moves in the positive direction