Derive the equation for kinetic energy

[Suraj M]

question:
I came across a question which said: Derive the equation for kinetic energy of a body of mass m and velocity v.
What do they mean?
Attempt:
I actually just wrote that by work energy theorem diff. in KE = work done
so Fx = KE
$$ max = m \frac {v²-u²}{2x} x = ½mv² - ½mu² $$
each of these terms represents a quantity called KE so this is the formula.
Is this right?? Because i seriously doubt it.

 

[PeroK]

question:
I came across a question which said: Derive the equation for kinetic energy of a body of mass m and velocity v.
What do they mean?
Attempt:
I actually just wrote that by work energy theorem diff. in KE = work done
so Fx = KE
$$ max = m \frac {v²-u²}{2x} x = ½mv² - ½mu² $$
each of these terms represents a quantity called KE so this is the formula.
Is this right?? Because i seriously doubt it.

That assumes constant acceleration. What if acceleration isn't constant?

 

[Suraj M]

Then i would do this:
$$ Fdx = dKE$$
$$ ma.dx = m. \frac{dv}{dt}dx = m v dv$$
$$mvdv = dKE$$
$$ m~₀∫^x vdv = ΔKE$$
$$ \frac{m(v^2 - u^2)}{2} = ΔKE $$