Question

Solve this derivation K.E=1/2mv2

Answer 1

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\red{\underline \bold{To \: Derive:}} \\ \tt: \implies K.E= \frac{1}{2} \: m {v}^{2}$

• According to given question :

$\text{Let \: a \:block \: of \: mass \: m \: with \: horizontaly\: applying} \\ \text{force \: F \: and \: travelling \: x \: distance \:attains \: a\: velocity \: v} \\ \\ \bold{As \: we \: know \: that } \\ \tt: \implies Work \: done = Force \times Displacement - - - - - (1) \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Force = Mass \times Acceleration - - - - - (2) \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies {v}^{2} = {u}^{2} + 2as \\ \\ \tt: \implies {v}^{2} = {0}^{2} + 2as \\ \\ \tt: \implies \frac{ {v}^{2} }{2s} = a \\ \\ \tt: \implies a= \frac{ {v}^{2} }{2x} - - - - - (3) \\ \\ \text{Putting \: value \: of \: (2) \: and \: (3) \: in \: (1)} \\ \\ \tt: \implies Work \: Done = m \times \frac{ {v}^{2} }{2x} \times x \\ \\ \tt: \implies Work \: Done = \frac{1}{2} m {v}^{2} \\ \\ \tt \circ \: Work \: Done = Kinetic \: Energy \\ \\ \green{\tt: \implies Kinetic \: Energy = \frac{1}{2} m {v}^{2} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \huge{\green{\text{Derived }}}$

Answer 2

$\large \bold \red{ \underline{ \underline{derivation \: of \: k.e = \frac{1}{2}mv {}^{2} }}} \\ \large \bold \green{ \underline{ \underline{explanation}}} \\ \large \bold{considerd \: the \: body \: is \: moving \: of \: \: mass \: m \: \: moving \: with \: velocity = v \: ms} \\ \\ \large \bold \blue{w =f.s} \\ \large \bold \blue{f = ma} \\ \large \bold \blue{w = m.a.s.......(1)} \\ \large \bold \orange{ \underline{ \underline{by \: third \: equation \: of \: motion}}} \\ \large \bold \green{v {}^{2} = {u}^{2} + 2as } \\ \large \bold{here \: u = 0 \: \: since \: the \: body \: is \: initially \: rest} \\ \large \bold \green{v {}^{2} = 2as } \\ \large \bold \green{as = \frac{1}{2} {v}^{2} } \\ \large \bold{substitute \: above \: value \: in \: equation \: (1)} \\ \large \bold \green{w = \frac{1}{2}m {v}^{2} } \\ \large \bold \red{ \underline {\underline{k.e = \frac{1}{2}m {v}^{2} = proved }}}$