The displacement of a particle starting from rest (at `t = 0)` is given by `s = 6t^(2) - t^(3)`. The time in seconds at which the particle will attain zero velocity again, is

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the time at which the particle attains zero velocity again after starting from rest. The displacement of the particle is given by the equation:

\[ s = 6t^2 - t^3 \]

### Step 1: Find the velocity of the particle
The velocity \( v \) is the first derivative of the displacement \( s \) with respect to time \( t \):

\[ v = \frac{ds}{dt} = \frac{d}{dt}(6t^2 - t^3) \]

### Step 2: Differentiate the displacement equation
Using the power rule for differentiation:

\[ v = 12t - 3t^2 \]

### Step 3: Set the velocity equation to zero
To find when the particle attains zero velocity, we set the velocity equation to zero:

\[ 12t - 3t^2 = 0 \]

### Step 4: Factor the equation
We can factor out \( 3t \):

\[ 3t(4 - t) = 0 \]

### Step 5: Solve for \( t \)
Setting each factor to zero gives us the possible solutions:

1. \( 3t = 0 \) → \( t = 0 \)
2. \( 4 - t = 0 \) → \( t = 4 \)

### Step 6: Identify the times when velocity is zero
The particle starts from rest at \( t = 0 \) and attains zero velocity again at \( t = 4 \) seconds.

### Final Answer
The time at which the particle will attain zero velocity again is:

\[ \boxed{4 \text{ seconds}} \]

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