A particle of mass 2 kg travels according to `S=3t^3+2t` Find the work done in 3 second of time

Options

A. 6889 J
B. 6885 J
C. 1782 J
D. None of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given data
We are given:
- Mass of the particle (m) = 2 kg
- Displacement equation (S) = 3t³ + 2t
- Time (t) = 3 seconds

### Step 2: Find the velocity of the particle
Velocity (v) is the derivative of displacement (S) with respect to time (t).

Using the displacement equation:
\[ S = 3t^3 + 2t \]

We differentiate S with respect to t:
\[ v = \frac{dS}{dt} = \frac{d}{dt}(3t^3 + 2t) \]
\[ v = 9t^2 + 2 \]

### Step 3: Calculate the final velocity at t = 3 seconds
Now we substitute t = 3 seconds into the velocity equation:
\[ v_{final} = 9(3^2) + 2 \]
\[ v_{final} = 9(9) + 2 \]
\[ v_{final} = 81 + 2 \]
\[ v_{final} = 83 \, \text{m/s} \]

### Step 4: Calculate the initial velocity at t = 0 seconds
Now we calculate the initial velocity at t = 0 seconds:
\[ v_{initial} = 9(0^2) + 2 \]
\[ v_{initial} = 0 + 2 \]
\[ v_{initial} = 2 \, \text{m/s} \]

### Step 5: Use the work-energy theorem to find the work done
According to the work-energy theorem:
\[ W = \frac{1}{2} m (v_{final}^2 - v_{initial}^2) \]

Substituting the values:
\[ W = \frac{1}{2} \times 2 \times (83^2 - 2^2) \]
\[ W = 1 \times (6889 - 4) \]
\[ W = 1 \times 6885 \]
\[ W = 6885 \, \text{Joules} \]

### Step 6: Convert the work done into kilojoules
To convert Joules to kilojoules:
\[ W = \frac{6885}{1000} = 6.885 \, \text{kJ} \]

### Final Answer
The work done in 3 seconds is **6885 Joules** or **6.885 kJ**.

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