Question

derive the equation for kinetic energy

Answer 1

From equations of motion, the relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, s is  

But from Newton’s second law of motion F = m a.

work done by the force, F is written as the following:

If the object is starting from its stationary position, that is, u = 0, then

From work and energy theorem, work done is equal to the change in the kinetic energy of the object.

w = (1/2) mv 2

If u = 0, the work done will be,

Thus the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is

K.E = (1/2) mv 2

Answer 2

$\mathfrak{Derivation \: of \:Kinetic \: Energy}$

$\mathsf{Things \: to \: know \: before\: Derivation}$

$\mathsf{\implies Work done = Fs}$

$\mathsf{\implies v² = u² + 2as}$

$\mathsf{\implies s = \frac{v² - u²}{2a}}$

$\mathsf{\implies u = 0 m/s \: for \:a \: body \: starting \: from \: rest}$

$\mathsf{\implies Work \: Done = \: Energy}$

$\mathsf{\implies Kinetic \: Energy \: is \: also \: written \: as \: E_k}$

$\mathsf{\implies Force = mass \: × \: acceleration \: = ma}$

$\mathsf{Derivation}$

$\mathsf{\hookrightarrow E_k = Work done = Fs }$

$\mathsf{\hookrightarrow \: = \: Fs \: = ma × s }$

$\mathsf{\hookrightarrow E_k = m × \frac{v² - u²}{2a} × a}$

$\mathsf{\hookrightarrow E_k = \frac{1}{2}mv²}$