description
 WorkEnergy Theorem
notes
Notions of Work and Kinetic Energy
“The workenergy theorem states that the change in kinetic energy of a particle is equal to the work done on it by the net force.”

Kf – Ki = W
where Kf is the final Kinetic Energy and Ki is the initial Kinetic Energy 
We know the equation in 3D: v^{2} – u^{2} = 2 a.d (where u  initial velocity, v  final velocity, a acceleration, d  displacement)
Now multiplying the equation by m/2 we have,½ mv^{2} – ½ mu^{2} = ma.d
½ mv^{2} – ½ mu^{2} = F.d (Since ma = F)
The WorkEnergy Theorem for a Variable Force
The time rate for change of kinetic energy
`(dK)/(dt)=d/(dt)(1/2 mv^2)`
`=m(dv)/(dt)v`
`=Fv` ...(from Newton's Second Law)
`=F(dx)/(dt)`
Thus,
`dK=F dx`
Integrating from initial position (x_{i}) to final position (x_{f}), we have
`∫_(K_i)^(K_f) dK=∫_(x_i)^(x_f)F dx`
where, K_{t} and K_{f} are the initial and final kinetic energies corresponding to x_{i} and x_{f},
or `K_fK_i=∫_(x_i)^(x_f) F dx`
From earlier proved equation, it follows that
`K_fK_i =W`
Thus, the W.E. theorem is proved for a variable force.
Regarding the workenergy theorem it is worth noting that
(i) If W_{net} is positive, then K_{f} – K_{i} = positive, i.e., K_{f} > K_{i} or kinetic energy will increase and viceversa.
(ii) This theorem can be applied to noninertial frames also. In a noninertial frame it can be written as:
Work done by all the forces (including the Pseudo force) = change in kinetic energy in noninertial frame.