Derivation of Equations of Motion
The equations of motion describe the relationship between an object's position, velocity, acceleration, and time. These equations are derived based on the fundamental principles of physics. Here is a step-by-step guide to derive the equations of motion:

First Equation of Motion
- The first equation of motion relates an object's initial velocity, final velocity, acceleration, and displacement over a certain time period.
- It is derived from the equation of motion: \(v = u + at\), where \(v\) is final velocity, \(u\) is initial velocity, \(a\) is acceleration, and \(t\) is time.
- To derive this equation, we can start with the definition of acceleration: \(a = \frac{v-u}{t}\).
- Rearranging the equation gives us the first equation of motion: \(v = u + at\).

Second Equation of Motion
- The second equation of motion relates an object's displacement, initial velocity, acceleration, and time.
- It is derived from the equation of motion: \(s = ut + \frac{1}{2}at^2\), where \(s\) is displacement.
- To derive this equation, we can start with the definition of average velocity: \(v_{avg} = \frac{u+v}{2}\).
- Substituting the expression for average velocity into the equation gives us the second equation of motion: \(s = v_{avg} t\).

Third Equation of Motion
- The third equation of motion relates an object's final velocity, initial velocity, acceleration, and displacement.
- It is derived from the equation of motion: \(v^2 = u^2 + 2as\), where \(s\) is displacement.
- To derive this equation, we can start with the definition of acceleration: \(a = \frac{v-u}{t}\).
- Substituting the expression for acceleration into the equation gives us the third equation of motion: \(v^2 = u^2 + 2as\).
By following these steps and principles of physics, we can derive the equations of motion that describe the motion of objects in a given system.