The velocity of a particle increases from u to v in a time t during which it covers a distances S. If the particle has a uniform accerelation, which one of the following equations does not apply to the motion?

Options

A. 2S=(v+u)t
B. `a=(v-u)/(t)`
C. `v^(2) = u^(2)-2aS`
D. `S=(u+(1)/(2)at)t`

Explanation

To solve the problem, we need to identify which equation of motion does not apply to a particle moving with uniform acceleration, given that its velocity changes from \( u \) to \( v \) over a time \( t \) while covering a distance \( S \).

### Step-by-Step Solution:

1. **Understanding the Variables**:
- Initial velocity: \( u \)
- Final velocity: \( v \)
- Time taken: \( t \)
- Distance covered: \( S \)
- Uniform acceleration: \( a \)

2. **Using the First Equation of Motion**:
The first equation of motion relates final velocity, initial velocity, acceleration, and time:
\[ v = u + at \]
From this, we can express acceleration as:
\[ a = \frac{v - u}{t} \]

3. **Using the Second Equation of Motion**:
The second equation of motion relates distance, initial velocity, time, and acceleration:
\[ S = ut + \frac{1}{2}at^2 \]
Substituting \( a \) from the first equation into this gives:
\[ S = ut + \frac{1}{2} \left(\frac{v - u}{t}\right)t^2 \]
Simplifying this, we get:
\[ S = ut + \frac{1}{2}(v - u)t \]
Rearranging yields:
\[ 2S = (u + v)t \]

4. **Using the Third Equation of Motion**:
The third equation of motion relates the velocities and distance:
\[ v^2 = u^2 + 2aS \]
Substituting \( a \) from the first equation gives:
\[ v^2 = u^2 + 2\left(\frac{v - u}{t}\right)S \]

5. **Identifying the Incorrect Equation**:
The options provided include:
- \( v^2 = u^2 + 2aS \)
- \( v^2 = u^2 - 2S \) (This option is incorrect because it suggests a negative relationship which does not apply in this context)
- \( S = ut + \frac{1}{2}at^2 \)
- \( 2S = (u + v)t \)

The equation \( v^2 = u^2 - 2S \) is not applicable because it does not correctly represent the relationship between the variables when acceleration is positive. The correct form should be \( v^2 = u^2 + 2aS \).

### Conclusion:
The equation that does not apply to the motion is:
\[ \text{Option C: } v^2 = u^2 - 2S \]