(a)

The function is .

Where is mass of the object , the velocity is  and position of the object is  .

Find and at anytime and the total distance.

Consider  .

Since .

Integrate on both sides.

Take exponential on both sides.

Where is the initial velocity of the function.

The velocity of the function is .

.

Integrate both sides with respect to

Since .

At the initial condition .

Since .

Substitute in .

Therefore, .

The total distance is .

Substitute .

.

Therefore ,the total distance is .

(b)

The function is .

Where is mass of the object , the velocity is  and position of the object is  .

Find and at anytime and the total distance.

Consider  .

Since .

Integrate on both sides.

At the initial condition  and .

Therefore, the velocity is .

.

Integrate both sides with respect to

Since .

At the initial condition .

Since .

Substitute  in .

Therefore, .

The total distance is .

Substitute .

.

Therefore, the total distance is infinity.

(a)  , . \ \

The total distnace is .

(b) , .

The total distance is infinity.