Question

prove that KE= ½mv²

Answer 1

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In the attachment..

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Answer 2

              Kinetic energy =  $\frac{1}{2}\ mv^{2}$

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Let initial velocity = u

Let applied force = F.

Acceleration = a

Time = t

Displacement during time = s

Final velocity = v

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According to the 2nd law of motion,

F = ma -------- (i)    (Work done on an object W = F * s)

By using 2nd equation of motion,

S = ut + 1/2 at²  (But, Initial velocity is 0)

S = 0 + 1/2 at²

Therefore,

S = 1/2 at² ------------ (ii)

Adding equation (i) and (ii)

W = ma * 1/2 at²

∴ W = 1/2 m(at)² ------------- (iii)

Now, By using first equation of Motion,

V = u + at [ (U) Initial velocity is 0]

V = 0 + at

V = at

∴ V² = at² -------------- (iv)

From equation (iii) and (iv)

W = 1/2 mv²

We know that, Work done by an object is equal to it's Kinetic Energy.

i.e. K.E = W

Therefore,

                             Kinetic Energy = 1/2 mv²