Maximum Velocity attained by the car:
To find the maximum velocity attained by the car, we need to determine the time at which the car reaches its maximum velocity.

Let's assume that the car accelerates for a time period of t1 seconds before decelerating. During this time, the car covers a distance given by the equation:

S1 = (1/2) * a * (t1)^2

Now, let's calculate the time t2 when the car starts decelerating. Since the car started from rest, the distance covered during acceleration is the same as the distance covered during deceleration. Therefore, the distance covered during deceleration is also S1.

Using the equation of motion:

S = ut + (1/2) * a * t^2

where S represents the distance covered, u represents the initial velocity, a represents the acceleration, and t represents the time,

we can write the equation for deceleration as:

S1 = (1/2) * b * (t - t2)^2

Since the car comes to rest during deceleration, the final velocity at the end of deceleration is 0. Therefore, we have:

0 = b * (t - t2)

From this equation, we can find the value of t2 as:

t2 = t - (S1 / b)^0.5

Now, to find the maximum velocity, we need to consider the time period during which the car is decelerating. The initial velocity during deceleration is the maximum velocity attained during acceleration. Therefore, the maximum velocity is given by:

v_max = a * t1

Total Distance travelled by the car:
To find the total distance travelled by the car, we need to sum up the distances covered during acceleration and deceleration.

The distance covered during acceleration is given by S1 = (1/2) * a * (t1)^2.

The distance covered during deceleration is given by S2 = (1/2) * b * (t - t2)^2.

Therefore, the total distance travelled by the car is:

S_total = S1 + S2

Substituting the values of S1 and t2, we get:

S_total = (1/2) * a * (t1)^2 + (1/2) * b * (t - t2)^2

Simplifying further, we get:

S_total = (1/2) * a * (t1)^2 + (1/2) * b * (t - (S1 / b)^0.5)^2

Therefore, the total distance travelled by the car is given by this equation in terms of a, b, and t.

In summary, the maximum velocity attained by the car is v_max = a * t1, and the total distance travelled by the car is S_total = (1/2) * a * (t1)^2 + (1/2) * b * (t - (S1 / b)^0.5)^2, where S1 = (1/2) * a * (t1)^2, and t1 = t - (S1 / a)^0.5.