#### To determine

**(a)**

**To find:**

We have to show that the acceleration is proportional to the velocity.

#### Answer

**Answer: a∝v**

#### Explanation

**Calculation**:

The velocity of a particle that moves in a straight line under the influence of viscous forces is vt=C e-kt.

First, we find the acceleration of a particle as

a=dvdt=-kCe-kt.

Since, C and k are both positive constants. So, we say k1=-Ck. So

a=k1e-kt=k1v

Therefore

a∝v

**Conclusion: a∝v**

#### To determine

**(b)**

**To find:** The initial velocity

#### Answer

C is the initial velocity.

#### Explanation

Now the velocity of particle at t=0 or initial velocity of a particle is obtained by putting t=0.

So, v0=Ce0=C_{.}

Therefore, C is the initial velocity.

#### To determine

**(C)**

**To find-** We have to find the time when velocity equal to half the initial velocity.

#### Answer

at time -1kln12, the velocity is half of the initial velocity.

#### Explanation

**Calculation-** Initial velocity is obtained by putting t=0 in vt=Ce-kt. So, after putting t=0, we get

v0=C

Therefore, the initial velocity is C.

Let t1 be the time when the velocity v(t1) equals the half of the initial velocity C.

So,

vt1=C2

Or

Ce-kt1=C2

Or e-kt1=12

Now taking natural logarithmic function on both sides, we have

lne-kt1=ln12

Therefore,

-kt1=ln12

Or t1=-1kln12

Thus, at time -1kln12, the velocity is half of the initial velocity.

**Conclusion:** at time -1kln12, the velocity is half of the initial velocity.