Equation `s_t=u + at - 1/2 a` does not seem dimensionally correct, why?

Explanation

To analyze the equation \( s_t = u + at - \frac{1}{2} a \) for dimensional correctness, we will break it down step by step.

### Step 1: Identify the dimensions of each term in the equation

1. **Dimension of \( s_t \)**:
- \( s_t \) represents displacement, which has the dimension of length (L).
- Dimension: \( [s_t] = L \)

2. **Dimension of \( u \)**:
- \( u \) represents initial velocity, which has the dimension of length per time (L/T).
- Dimension: \( [u] = L T^{-1} \)

3. **Dimension of \( at \)**:
- \( a \) represents acceleration, which has the dimension of length per time squared (L/T²).
- \( t \) represents time, which has the dimension of time (T).
- Therefore, the dimension of \( at \) is:
\[ [at] = [a][t] = (L T^{-2})(T) = L T^{-1} \]

4. **Dimension of \( -\frac{1}{2} a \)**:
- This term is a bit misleading because it seems to suggest that we are subtracting acceleration directly. However, we need to consider the context of the equation.
- If we consider it as \( -\frac{1}{2} a t^2 \), we can analyze it as follows:
- \( a \) has the dimension \( L T^{-2} \) and \( t^2 \) has the dimension \( T^2 \).
- Therefore, the dimension of this term is:
\[ \left[-\frac{1}{2} a t^2\right] = [a][t^2] = (L T^{-2})(T^2) = L \]

### Step 2: Combine the dimensions of all terms

Now, we can rewrite the equation considering the proper dimensions:

\[ s_t = u + at - \frac{1}{2} a t^2 \]

- The dimensions of \( u \) and \( at \) are both \( L T^{-1} \).
- The dimension of \( -\frac{1}{2} a t^2 \) is \( L \).

### Step 3: Check for dimensional consistency

In the equation \( s_t = u + at - \frac{1}{2} a t^2 \):

- The left-hand side \( s_t \) has the dimension \( L \).
- The right-hand side consists of:
- \( u \) has dimension \( L T^{-1} \)
- \( at \) has dimension \( L T^{-1} \)
- \( -\frac{1}{2} a t^2 \) has dimension \( L \)

Since \( u \) and \( at \) have dimensions of \( L T^{-1} \), they cannot be added to \( -\frac{1}{2} a t^2 \) which has dimension \( L \). Therefore, the equation as presented is not dimensionally consistent.

### Conclusion

The equation \( s_t = u + at - \frac{1}{2} a \) is not dimensionally correct because it combines terms of different dimensions (length and length per time).