Displacement (S) of a particle is given as `S = (t^3 – t^2)`. Find the speed of the particle when it's acceleration is zero

Options

A. 1 m/s
B. 2 m/s
C. `-1/3` m/s
D. `2/3` m/s

Explanation

To solve the problem step by step, we will follow these procedures:

### Step 1: Write the displacement equation
The displacement \( S \) of the particle is given by:
\[ S = t^3 - t^2 \]

### Step 2: Differentiate to find velocity
To find the velocity \( V \), we differentiate the displacement \( S \) with respect to time \( t \):
\[ V = \frac{dS}{dt} = \frac{d}{dt}(t^3 - t^2) \]
Using the power rule of differentiation:
\[ V = 3t^2 - 2t \]

### Step 3: Differentiate to find acceleration
Next, we differentiate the velocity \( V \) to find the acceleration \( A \):
\[ A = \frac{dV}{dt} = \frac{d}{dt}(3t^2 - 2t) \]
Again using the power rule:
\[ A = 6t - 2 \]

### Step 4: Set acceleration to zero
We need to find the time \( t \) when the acceleration is zero:
\[ 6t - 2 = 0 \]
Solving for \( t \):
\[ 6t = 2 \implies t = \frac{1}{3} \text{ seconds} \]

### Step 5: Substitute time back to find velocity
Now, we substitute \( t = \frac{1}{3} \) back into the velocity equation to find the speed of the particle:
\[ V = 3\left(\frac{1}{3}\right)^2 - 2\left(\frac{1}{3}\right) \]
Calculating this:
\[ V = 3 \cdot \frac{1}{9} - \frac{2}{3} = \frac{1}{3} - \frac{2}{3} = -\frac{1}{3} \text{ m/s} \]

### Final Answer
The speed of the particle when its acceleration is zero is:
\[ \text{Speed} = -\frac{1}{3} \text{ m/s} \]