https://file.aiquickdraw.com/mathbot/file/1726664804350vcw7w6ov.jpg

Let's begin by deriving the two kinematic equations shown in the image:

### (1) Derivation of \( v_f = v_i + at \)

This is a standard kinematic equation, where:
- \( v_f \) is the final velocity,
- \( v_i \) is the initial velocity,
- \( a \) is the acceleration, and
- \( t \) is the time.

**Step-by-step derivation:**

1. **Definition of acceleration**: Acceleration is defined as the rate of change of velocity.
   \[
   a = \frac{v_f - v_i}{t}
   \]
   
2. **Rearranging for \( v_f \)**: Multiply both sides by \( t \) to isolate \( v_f \).
   \[
   a t = v_f - v_i
   \]
   
3. **Solving for \( v_f \)**:
   \[
   v_f = v_i + at
   \]

Thus, we derive:
\[
v_f = v_i + at
\]

### (2) Derivation of \( \Delta x = \left( \frac{v_f + v_i}{2} \right) t \)

This is another common kinematic equation, representing the displacement (\( \Delta x \)).

**Step-by-step derivation:**

1. **Average velocity**: The average velocity is defined as:
   \[
   v_{\text{avg}} = \frac{v_f + v_i}{2}
   \]
   
2. **Relationship between displacement and average velocity**: Displacement is the product of average velocity and time:
   \[
   \Delta x = v_{\text{avg}} \cdot t
   \]
   
3. **Substitute the expression for average velocity**:
   \[
   \Delta x = \left( \frac{v_f + v_i}{2} \right) t
   \]

Thus, we derive:
\[
\Delta x = \left( \frac{v_f + v_i}{2} \right) t
\]

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Would you like further details or have any questions? Here are five related questions that could deepen your understanding:

1. How can these equations be applied to motion with constant acceleration?
2. What would happen to these equations if the object was in free fall (where \( a = g \))?
3. How do these equations change if the acceleration is zero?
4. Can you derive a third kinematic equation that eliminates time from the expressions?
5. How do we handle negative values for acceleration or velocity?

**Tip**: Always ensure units are consistent when solving kinematic problems (e.g., meters for distance, seconds for time).

Derive the kinematic equations (1) vf = vi + at and (2) Δx = ((vf + vi)/2)t.

Explore the step-by-step derivation of two key kinematic equations: (1) vf = vi + at, which relates final velocity to initial velocity, acceleration, and time, and (2) Δx = ((vf + vi)/2)t, representing displacement using average velocity. These formulas are essential for understanding linear motion in physics, suitable for students in Grades 9-12.

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