Deduce the relation for work- Energy Theorem.

Explanation

(i) It states that work done by the force acting on a body is equal to the change produced in the kinetic energy of the body.
(ii) Consider a body of mass m at rest on a frictionless horizontal surface.
(iii) The work (W) done by the constant force (F) for a displacement (s) in the same direction is,
`W=Fs" "...(1)`
The constant force is given by the equation,
`F=ma" "...(2)`
The third equation of motion can be written as,
`v^(2)=u^(2)+2as`
`a=(v^(2)-u^(2))/(2s)" "...(3)`
Substituting for a in equation (2),
`F=m((v^(2)-u^(2))/(2s))" "...(4)`
Substituting equation (4) in (1),
`W=m((v^(2))/(2s)s)-m((u^(2))/(2s)s)`
`W=(1)/(2)mv^(2)-(1)/(2)m u^(2)" "...(5)`
The expression for kinetic energy:
(i) The term `((1)/(2)mv^(2))` in t equation (5) is the kinetic energy of the body of mass (m) moving with velocity (v).
`KE=(1)/(2)mv^(2)" "...(6)`
(ii) Kinetic energy of the bodyis always positive. From equations (5) and (6)
`Delta KE=(1)/(2)mv^(2)-(1)/(2)m u^(2)" "...(7)`
Thus, `W=DeltaKE`
(iii) The expression on the right hand side (RHS) of equation (7) is the change in kinetic energy `(DeltaKE)` of the body.
(iv) This implies that the work done by the force on the body changes the kinetic energy of the body. This is called work-kinetic energy theorem.