Assertion: The given equation `x = x_(0) + u_(0)t + (1)/(2) at^(2)` is dimensionsally correct, where x is the distance travelled by a particle in time t , initial position `x_(0) ` initial velocity `u_(0)` and uniform acceleration a is along the direction of motion.
Reason: Dimensional analysis can be used for cheking the dimensional consistency or homogenetly of the equation.

Options

A. if both assertion and reason are true reason is the correct explanation of assertion.
B. If both assertion and reason are true but reason is not the correct explanation fo assertion.
C. If assertion is true but reaso is false.
D. IF both assertion and reason are false.

Explanation

To solve the given problem, we need to analyze the assertion and reason provided and check the dimensional correctness of the equation \( x = x_0 + u_0 t + \frac{1}{2} a t^2 \).

### Step-by-Step Solution:

1. **Identify the Left-Hand Side (LHS)**:
The LHS of the equation is \( x \), which represents the distance traveled by a particle. The dimension of distance is denoted as:
\[ [x] = L \]

2. **Identify the Right-Hand Side (RHS)**:
The RHS consists of three terms: \( x_0 \), \( u_0 t \), and \( \frac{1}{2} a t^2 \).

- **First Term**: \( x_0 \) is the initial position, which is also a distance. Thus, its dimension is:
\[ [x_0] = L \]

- **Second Term**: \( u_0 t \) is the product of initial velocity \( u_0 \) and time \( t \). The dimension of initial velocity is:
\[ [u_0] = L T^{-1} \]
Therefore, the dimension of \( u_0 t \) is:
\[ [u_0 t] = [u_0][t] = (L T^{-1})(T) = L \]

- **Third Term**: \( \frac{1}{2} a t^2 \) involves acceleration \( a \) and time \( t \). The dimension of acceleration is:
\[ [a] = L T^{-2} \]
Thus, the dimension of \( a t^2 \) is:
\[ [a t^2] = [a][t^2] = (L T^{-2})(T^2) = L \]

3. **Combine the RHS Terms**:
Now, we can combine the dimensions of all terms on the RHS:
\[ [RHS] = [x_0] + [u_0 t] + \left[\frac{1}{2} a t^2\right] = L + L + L = L \]

4. **Check Dimensional Consistency**:
Now we compare the dimensions of the LHS and RHS:
\[ [LHS] = L \quad \text{and} \quad [RHS] = L \]
Since both sides have the same dimension, the equation is dimensionally correct.

5. **Evaluate the Reason**:
The reason states that dimensional analysis can be used for checking the dimensional consistency or homogeneity of the equation. This is indeed true, as we have demonstrated that both sides of the equation have the same dimensions.

6. **Conclusion**:
Since both the assertion and reason are true, and the reason correctly explains the assertion, the correct option is:
**Option 1**: Both assertion and reason are true, and the reason is the correct explanation of the assertion.