Question

derivate kinetic eqution​

Answer 1

Answer:

Expression for  Kinetic energy(Ek)-

Consider an object with mass m, moving with initial velocity u. Let a force F, act on it causing a displacement s, along the direction of force and accelaration. Let the object attain final velocity v.

We know F = m x a ------------------------ 1

By position velocity relation we have,

V2 = u2 + 2as

or s = v2 -u2 / 2a ----------------------------- 2

The work done by the force

W = F x s -------------------------------------- 3

Substituting 1 and 2 in 3

W = (m x a) x (v2 - u2/2a)

W = m/2 (v2 - u2)

W = 1/2 mv2 - 1/2 mu2 -------------------- 4

For an object to be at rest in the beggining, u = 0

Therefore, W = 1/2 mv2

W = Ek ( Ek = kinetic energy)

Therefore, Ek = 1/2 mv2

Answer 2

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

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

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

$\mathsf{\implies v² = u² + 2as}$

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

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

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

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

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

$\mathsf{Derivation}$

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

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

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

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