Write the first equation of motion ?

Answer

To derive the first equation of motion, we will consider the concepts of velocity, acceleration, and time. Here’s a step-by-step solution:

### Step-by-Step Solution

1. **Understanding the Variables**:
- Let \( u \) be the initial velocity of the object.
- Let \( v \) be the final velocity of the object.
- Let \( a \) be the acceleration of the object.
- Let \( t \) be the time taken for the change in velocity.

2. **Definition of Acceleration**:
- Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, it can be expressed as:
\[ a = \frac{\Delta v}{\Delta t} \]
where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time.

3. **Expressing Change in Velocity**:
- The change in velocity \( \Delta v \) can be expressed as:
\[ \Delta v = v - u \]
where \( v \) is the final velocity and \( u \) is the initial velocity.

4. **Expressing Change in Time**:
- The change in time \( \Delta t \) can be expressed as:
\[ \Delta t = t \]
where \( t \) is the total time taken for the change.

5. **Substituting into the Acceleration Formula**:
- Substituting the expressions for \( \Delta v \) and \( \Delta t \) into the acceleration formula gives:
\[ a = \frac{v - u}{t} \]

6. **Rearranging the Equation**:
- Rearranging the equation to isolate \( v \) gives:
\[ v - u = at \]

7. **Final Equation**:
- Adding \( u \) to both sides results in the first equation of motion:
\[ v = u + at \]

### Final Answer
The first equation of motion is:
\[ v = u + at \]