Position (in m) of a particle moving on a straight line varies with time (in sec) as `x=t^(3)//3-3t^(2)+8t+4 (m)`. Consider the motion of the particle from t=0 to t=5 sec. `S_(1)` is the total distance travelled and `S_(2)` is the distance travelled during retardation. if `s_(1)//s_(2)=((3alpha+2))/11` the find `alpha`.

Explanation

To solve the problem step by step, we will analyze the motion of the particle given by the position function \( x(t) = \frac{t^3}{3} - 3t^2 + 8t + 4 \).

### Step 1: Find the velocity function
The velocity \( v(t) \) is the derivative of the position function \( x(t) \) with respect to time \( t \).

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}\left(\frac{t^3}{3} - 3t^2 + 8t + 4\right) \]
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