A particle is moving such that `s=t^(3)-6t^(2)+18t+9`, where s is in meters and t is in meters and t is in seconds. Find the minimum velocity attained by the particle.

Options

A. `29 ms^(-1)`
B. `5 ms^(-1)`
C. `6 ms^(-1)`
D. `12 ms^(-1)`

Explanation

To find the minimum velocity attained by the particle given the displacement function \( s(t) = t^3 - 6t^2 + 18t + 9 \), we will follow these steps:

### Step 1: Differentiate the displacement function to find the velocity function.
The velocity \( v(t) \) is the first derivative of the displacement \( s(t) \) with respect to time \( t \).

\[ v(t) = \frac{ds}{dt} = \frac{d}{dt}(t^3 - 6t^2 + 18t + 9) \]

### Step 2: Calculate the derivative.
Using the power rule of differentiation:

\[ v(t) = 3t^2 - 12t + 18 \]

### Step 3: Set the velocity function equal to zero to find critical points.
To find the minimum velocity, we need to find the critical points by setting \( v(t) = 0 \):

\[ 3t^2 - 12t + 18 = 0 \]

### Step 4: Simplify the equation.
Dividing the entire equation by 3 gives:

\[ t^2 - 4t + 6 = 0 \]

### Step 5: Use the quadratic formula to find the roots.
The quadratic formula is given by:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1, b = -4, c = 6 \):

\[ t = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} \]
\[ t = \frac{4 \pm \sqrt{16 - 24}}{2} \]
\[ t = \frac{4 \pm \sqrt{-8}}{2} \]

Since the discriminant is negative (\(-8\)), there are no real roots, which indicates that the velocity does not become zero at any real time \( t \).

### Step 6: Analyze the velocity function.
To find the minimum value of the velocity function, we can complete the square for the quadratic expression:

\[ v(t) = 3(t^2 - 4t + 6) \]
\[ = 3((t - 2)^2 + 2) \]

### Step 7: Determine the minimum velocity.
The expression \((t - 2)^2\) is always non-negative, and its minimum value is 0 when \( t = 2 \). Therefore, the minimum velocity occurs at:

\[ v(2) = 3(0 + 2) = 6 \text{ m/s} \]

### Conclusion:
The minimum velocity attained by the particle is \( 6 \text{ m/s} \).

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