Derivation of Equations of Motion

First Equation

Starting from:

$$ a = \frac{dv}{dt} $$

Assuming constant acceleration:

$$ dv = a \, dt $$

Integrating:

$$ \int dv = \int a \, dt $$

$$ v = u + at $$

Second Equation

We know:

$$ v = \frac{dx}{dt} $$

Substitute $v = u + at$:

$$ \frac{dx}{dt} = u + at $$

Integrating:

$$ x = ut + \frac{1}{2}at^2 $$

Third Equation

From:

$$ v = u + at $$

Eliminate $t$:

$$ t = \frac{v - u}{a} $$

Substitute into second equation:

$$ x = ut + \frac{1}{2}at^2 $$

After simplification:

$$ v^2 = u^2 + 2ax $$