The distance travelled by a particle starting from rest and moving with an acceleration `(4)/(3) ms^-2`, in the third second is.

Options

A. 6m
B. 4m
C. `(10)/(3)m`
D. `(19)/(3)m`

Answer

To find the distance traveled by a particle starting from rest and moving with an acceleration of \( \frac{4}{3} \, \text{m/s}^2 \) in the third second, we can use the formula for the distance traveled in the nth second of motion:

\[ S_n = u + \frac{1}{2} a (2n - 1) \]

Where:
- \( S_n \) is the distance traveled in the nth second.
- \( u \) is the initial velocity (which is 0 since the particle starts from rest).
- \( a \) is the acceleration.
- \( n \) is the second for which we want to find the distance (in this case, \( n = 3 \)).

### Step-by-step Solution:

1. **Identify the given values:**
- Initial velocity, \( u = 0 \, \text{m/s} \)
- Acceleration, \( a = \frac{4}{3} \, \text{m/s}^2 \)
- We want to find the distance traveled in the third second, so \( n = 3 \).

2. **Substitute the values into the formula:**
\[ S_3 = 0 + \frac{1}{2} \times \frac{4}{3} \times (2 \times 3 - 1) \]

3. **Calculate \( 2n - 1 \):**
\[ 2 \times 3 - 1 = 6 - 1 = 5 \]

4. **Substitute back into the equation:**
\[ S_3 = \frac{1}{2} \times \frac{4}{3} \times 5 \]

5. **Multiply the terms:**
\[ S_3 = \frac{1}{2} \times \frac{4 \times 5}{3} = \frac{1}{2} \times \frac{20}{3} = \frac{20}{6} = \frac{10}{3} \, \text{m} \]

6. **Final result:**
The distance traveled by the particle in the third second is \( \frac{10}{3} \, \text{m} \).

### Answer:
The distance traveled by the particle in the third second is \( \frac{10}{3} \, \text{m} \).

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