The displacement of an object of mass 3 kg is given by the relation `S=(1)/(3)t^(2)`, where t is time in seconds. If the work done by the net force on the object in 2 s is `(p)/(q)` joule, where p and q are smallest integer values, then what is the value of `p+q`?

Answer

To solve the problem, we need to find the work done by the net force on an object of mass 3 kg, given its displacement as a function of time. The displacement is given by the equation:

\[ S = \frac{1}{3} t^2 \]

### Step 1: Find the velocity of the object
To find the velocity, we differentiate the displacement with respect to time:

\[ v = \frac{dS}{dt} = \frac{d}{dt} \left( \frac{1}{3} t^2 \right) \]

Calculating the derivative:

\[ v = \frac{2}{3} t \]

### Step 2: Calculate the velocity at t = 2 seconds
Now, we substitute \( t = 2 \) seconds into the velocity equation:

\[ v(2) = \frac{2}{3} \times 2 = \frac{4}{3} \, \text{m/s} \]

### Step 3: Calculate the initial velocity at t = 0 seconds
At \( t = 0 \):

\[ v(0) = \frac{2}{3} \times 0 = 0 \, \text{m/s} \]

### Step 4: Calculate the change in kinetic energy
The work done by the net force is equal to the change in kinetic energy, which can be calculated using the formula:

\[ \text{Work} = \Delta KE = KE_{\text{final}} - KE_{\text{initial}} \]

Where:

\[ KE = \frac{1}{2} m v^2 \]

Substituting the values:

- Mass \( m = 3 \, \text{kg} \)
- Final velocity \( v_{\text{final}} = \frac{4}{3} \, \text{m/s} \)
- Initial velocity \( v_{\text{initial}} = 0 \, \text{m/s} \)

Calculating the final kinetic energy:

\[ KE_{\text{final}} = \frac{1}{2} \times 3 \times \left( \frac{4}{3} \right)^2 \]

Calculating \( \left( \frac{4}{3} \right)^2 \):

\[ \left( \frac{4}{3} \right)^2 = \frac{16}{9} \]

Now substituting back:

\[ KE_{\text{final}} = \frac{1}{2} \times 3 \times \frac{16}{9} = \frac{48}{18} = \frac{8}{3} \, \text{J} \]

Since the initial kinetic energy is zero:

\[ KE_{\text{initial}} = 0 \]

Thus, the work done is:

\[ \text{Work} = \frac{8}{3} - 0 = \frac{8}{3} \, \text{J} \]

### Step 5: Identify p and q
From the work done, we have:

\[ \text{Work} = \frac{p}{q} = \frac{8}{3} \]

Here, \( p = 8 \) and \( q = 3 \).

### Step 6: Calculate \( p + q \)
Finally, we calculate:

\[ p + q = 8 + 3 = 11 \]

Thus, the answer is:

**Final Answer: 11**