To derive the equations of uniformly accelerated motion, we'll use the basic principles of kinematics. Here are the three key equations for uniformly accelerated motion, along with their derivations:
### 1. Equation: \( v = u + at \)
**Derivation:**
- Let \( u \) be the initial velocity.
- Let \( v \) be the final velocity.
- Let \( a \) be the acceleration.
- Let \( t \) be the time.
Acceleration is defined as the rate of change of velocity. Therefore, we can express acceleration as:
\[ a = \frac{v - u}{t} \]
Rearranging this equation to solve for \( v \):
\[ v = u + at \]
### 2. Equation: \( s = ut + \frac{1}{2}at^2 \)
**Derivation:**
- Let \( s \) be the displacement.
To derive this equation, we use the fact that displacement \( s \) is the area under the velocity-time graph, which forms a trapezoid in the case of constant acceleration.
1. **Average Velocity**: The average velocity \( \bar{v} \) for uniformly accelerated motion is given by:
\[ \bar{v} = \frac{u + v}{2} \]
2. **Displacement**: The displacement \( s \) is the average velocity multiplied by time:
\[ s = \bar{v} \cdot t \]
Substitute \( \bar{v} \) into the displacement formula:
\[ s = \frac{u + v}{2} \cdot t \]
Substitute \( v = u + at \) into the equation:
\[ s = \frac{u + (u + at)}{2} \cdot t \]
\[ s = \frac{2u + at}{2} \cdot t \]
\[ s = (u + \frac{1}{2}at) \cdot t \]
\[ s = ut + \frac{1}{2}at^2 \]
### 3. Equation: \( v^2 = u^2 + 2as \)
**Derivation:**
To derive this equation, we start with the two previously derived equations.
1. From \( v = u + at \), solve for \( t \):
\[ t = \frac{v - u}{a} \]
2. Substitute \( t \) into the displacement equation \( s = ut + \frac{1}{2}at^2 \):
\[ s = u \left(\frac{v - u}{a}\right) + \frac{1}{2} a \left(\frac{v - u}{a}\right)^2 \]
Simplify:
\[ s = \frac{u(v - u)}{a} + \frac{1}{2} \cdot \frac{a(v - u)^2}{a^2} \]
\[ s = \frac{u(v - u)}{a} + \frac{1}{2} \cdot \frac{(v - u)^2}{a} \]
\[ s = \frac{u(v - u) + \frac{1}{2} (v^2 - 2uv + u^2)}{a} \]
\[ s = \frac{v^2 - u^2}{2a} \]
Rearrange to solve for \( v^2 \):
\[ v^2 = u^2 + 2as \]
These three equations are fundamental in kinematics for analyzing uniformly accelerated motion.
