Question

Derive an expression for the kinetic energy of a body by the calculus method.

Answer 1

as we know,

$usingconvservationofenergy$

$W.D.=\Delta KE?$

$asweknowdw=\int F.dx$

$=\int ma.dx$

$=m\int v\frac{dv}{dx}dxcos{0}^{\theta }$

$=m{\int }_u^{v}vdv$

$=m{\left[\frac{v^2}{2}\right]}_u^v$

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

$=\Delta KE$

if body start from rest as we applied force, u=0

$\Delta KE=\frac{1}{2}m{v}^2$