Chek the correctness of the relation, `S_(nth) = u +(a)/(2)(2n -1),` where u is initial velocity, a is acceleratin and `S_(nth)` is the distance travelled by the body in nth second.

Options

A. `M^(0)L^(1)T^(0)`
B. `M^(0)L^(-1)T^(1)`
C. `M^(0)L^(1)T^(-1)`
D. `M^(0)L^(0)T^(1)`

Answer

To check the correctness of the relation \( S_{nth} = u + \frac{a}{2}(2n - 1) \), where \( u \) is the initial velocity, \( a \) is the acceleration, and \( S_{nth} \) is the distance traveled by the body in the nth second, we will analyze the dimensions of each term in the equation.

### Step-by-Step Solution:

1. **Identify the Dimensions of \( S_{nth} \)**:
- \( S_{nth} \) represents the distance traveled in the nth second.
- The dimension of distance is \( [L] \) (length).
- Therefore, the dimension of \( S_{nth} \) is \( [L] \).

2. **Identify the Dimensions of Initial Velocity \( u \)**:
- The dimension of velocity is given by distance/time.
- Hence, the dimension of \( u \) is \( [L][T^{-1}] \).

3. **Identify the Dimensions of Acceleration \( a \)**:
- The dimension of acceleration is given by change in velocity/time.
- Therefore, the dimension of \( a \) is \( [L][T^{-2}] \).

4. **Analyze the Term \( \frac{a}{2}(2n - 1) \)**:
- The term \( \frac{a}{2} \) has the same dimension as \( a \) since \( 2 \) is dimensionless.
- Thus, the dimension of \( \frac{a}{2} \) is also \( [L][T^{-2}] \).
- The term \( (2n - 1) \) is dimensionless because \( 2 \) and \( 1 \) are constants, and \( n \) represents the nth second, which has the dimension of time \( [T] \).
- Therefore, \( (2n - 1) \) does not contribute any additional dimensions and remains dimensionless.

5. **Combine the Dimensions**:
- Now, we need to find the dimension of the entire term \( \frac{a}{2}(2n - 1) \).
- Since \( (2n - 1) \) is dimensionless, the dimension of \( \frac{a}{2}(2n - 1) \) is the same as that of \( \frac{a}{2} \), which is \( [L][T^{-2}] \).

6. **Check the Overall Equation**:
- The equation is \( S_{nth} = u + \frac{a}{2}(2n - 1) \).
- The left-hand side \( S_{nth} \) has the dimension \( [L] \).
- The right-hand side consists of \( u \) (dimension \( [L][T^{-1}] \)) and \( \frac{a}{2}(2n - 1) \) (dimension \( [L][T^{-2}] \)).
- Since \( u \) and \( \frac{a}{2}(2n - 1) \) have different dimensions, the equation cannot be correct as it stands.

7. **Conclusion**:
- The dimensions on both sides of the equation do not match, indicating that the relation \( S_{nth} = u + \frac{a}{2}(2n - 1) \) is incorrect.