Question

Proved the Kinetic energy Ek1=1/2 mv²​

Answer 1

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The work done is accelerating an object is given by:

$\large W = F∆x$

Where $\large F$ is the force and $\large ∆x$ is the displacement.

If the object started from rest and all of the work was converted into Kinetic energy then this will be equal to the Kinetic energy of the object:

$\large K = F∆x$

Using Newton's second law,

$\large K = ma∆x = m(a∆x)$

Now using the equation of motion,

$\large 2a∆x = v^{2} - v^{2}_{0} → a∆x = \frac{v^{2}}{2} – \frac{v^{2}_{0}}{2}$

Substitute this into equation for kinetic energy to get:

$\large K = m( \frac{v^{2}}{2} – \frac{v^{2}_{0}}{2} )$

If the object started from rest then the initial velocity will be :

$\large v_{0} = 0$ so $\large K$ simplifies to:

$\large K = \frac{mv^{2}}{2}$

Hence, Proven.

Answer 2

Answer:

Proved the Kinetic energy.

$ek = \frac{1}{2}m {v}^{2}$

Let us consider of an object of mass m moving with a uniform velocity u .

Let it now be displaced through a distance as when a constant force f act on it in the direction of its displacement.

From the equation of work done we get ,

$ek = \frac{1}{2} m {v}^{2}$

$w = f \times s \: - - - - (i)$

As we know ,

F = ma ——— (ii)

and

2aS = v² – u²

=> S = v²–u²/ 2a ————(iii)

Putting in value of f and S in equation (i)

we get,

W = ma × v²–u²/2a

=½m(v²–u²)

Now, If the object starting from a stationary position i.e., U=0 , than

w=½mv²

But, it is clear that the work done is equal to the change in the Kinetic energy of an object. Thus , Kinetic energy will be —

Ek=½mv²