a car starts moving from rest with an acceleration whose value linearly increases with time from zero to 6m/s^2 in 6 sec after which it moves with constant velocity. find the time taken by the car to travel first 72m from starting point

To solve this problem, we need to break it down into two distinct phases: the acceleration phase and the constant velocity phase. Let's analyze each part step by step.

Phase 1: Acceleration

The car starts from rest and accelerates with a value that increases linearly from 0 to 6 m/s² over a period of 6 seconds. This means that the acceleration at any time \( t \) can be expressed as:

a(t) = kt

where \( k \) is the rate of increase of acceleration. Since the acceleration reaches 6 m/s² at \( t = 6 \) seconds, we can find \( k \) as follows:

6 = k * 6

From this, we find:

k = 1 m/s³

Now, the acceleration function becomes:

a(t) = t

To find the velocity as a function of time, we integrate the acceleration:

v(t) = ∫a(t) dt = ∫t dt = (1/2)t² + C

Since the car starts from rest, the constant \( C = 0 \). Thus:

v(t) = (1/2)t²

Next, we need to find the distance traveled during the acceleration phase. We can find the distance function by integrating the velocity:

s(t) = ∫v(t) dt = ∫(1/2)t² dt = (1/6)t³ + C

Again, since the car starts from the origin, \( C = 0 \). Therefore:

s(t) = (1/6)t³

Now, we need to determine the distance traveled in the first 6 seconds:

s(6) = (1/6)(6)³ = (1/6)(216) = 36 m

Phase 2: Constant Velocity

After 6 seconds, the car moves with a constant velocity. To find this velocity, we substitute \( t = 6 \) into the velocity equation:

v(6) = (1/2)(6)² = 18 m/s

Now, we need to find out how much more distance is required to reach a total of 72 m. Since the car has already traveled 36 m, the remaining distance is:

72 m - 36 m = 36 m

Now, we can calculate the time taken to cover this remaining distance at the constant velocity of 18 m/s:

Time = Distance / Velocity = 36 m / 18 m/s = 2 s

Final Calculation

The total time taken by the car to travel the first 72 m is the sum of the time taken during the acceleration phase and the time taken during the constant velocity phase:

Total Time = 6 s + 2 s = 8 s

In summary, the car takes a total of 8 seconds to travel the first 72 meters from the starting point.