Derive the formula for kinetic energy.

Question

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Answer 1

Let an object of mass m, move with uniform velocity u.
Let us displace it by s, due to constant force F, acting on it.
Work done on the object of mass 'm' is
$W = F \times s$ ...(i)
Due to the force, velocity changes to v and the acceleration produced is 'a'. Relationship between v, u, a and s can be given by formula $v^{2} - u^{2} = 2 a s$
$∴ s = \frac{v ^{2} - u ^{2}}{2 a}$ ..........(ii)
$F = m a$ ............(iii)
Substituting (ii) and (iii) in (i) we get
$W = F \times s$
$= m a \times \frac{v ^{2} - u ^{2}}{2 a}$
$W = \frac{1}{2} m (v^{2} - u^{2})$

If u=0,(object starts at rest)
$W = \frac{1}{2} m v^{2}$
Work done=Change in kinetic energy
$∴ E_{k} = \frac{1}{2} m v^{2}$

Answer 2

Let an object of mass m, move with uniform velocity u.

Let us displace it by s, due to constant force F, acting on it.

Work done on the object of mass 'm' is

W = F \times s

...(i)

Due to the force, velocity changes to v and the acceleration produced is 'a'. Relationship between v, u, a and s can be given by formula

v^{2} - u^{2} = 2 a s

∴ s = \frac{v ^{2} - u ^{2}}{2 a}

..........(ii)

F = m a

............(iii)

Substituting (ii) and (iii) in (i) we get

W = F \times s

= m a \times \frac{v ^{2} - u ^{2}}{2 a}

W = \frac{1}{2} m (v^{2} - u^{2})

If u=0,(object starts at rest)

W = \frac{1}{2} m v^{2}

Work done=Change in kinetic energy

∴ E_{k} = \frac{1}{2} m v^{2}