Question

Show that (a)KE=1/2mv2

Answer 1

`\boxed{ \bold{ \huge{ \boxed{ \sf{see \: below}}}}}`

Step-by-step explanation:

`\underline{ \bold{ \sf{To \: prove \: that \: kinetic \: energy = (1)/(2) m {v}^(2) }}}`

Let us consider, a body of mass ' m ' is lying at rest ( initial velocity = 0 ) on a smooth surface. Let a constant force F displaces this body in its own direction by a displacement ' d '. Let 'v' be it's final velocity. The work done ' W ' by the force is given by :

`\sf{W = FD}`

⇒
`\sf{W = m \: * a \: * s} \: \: \: \: \: \: \: \: \: ( \: ∴ \: f \: = \: ma \: ; \: s \: = d)`

⇒
`\sf{W = m \: * (v - u)/(t) * (u + v)/(2) * t \: \: \: \: \: \: \: \: \: (∴ \: a = (v - u)/(t) and \: s = (u + v)/(2) * t}`

⇒
`\sf{W = m * \frac{ {v}^(2) - {u}^(2) }{2} }`

⇒
`\sf{W = (1)/(2) m {v}^(2) \: \: \: \: \: \: \: \: \: \: \: \: (since, \: initial \: velocity(u) = 0)}`

The work done becomes the kinetic energy of the body. Thus, the kinetic energy of a body of mass ' m : moving with the velocity equal to 'v ' is 1 / 2 mv²

∴
`\sf{KE= (1)/(2) m {v}^(2) }`

`\sf{ \underline{ \bold{ {proved}}}}`

Hope I helped!

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