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# Notions of Work and Kinetic Energy: the Work-Energy Theorem

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• Work-Energy Theorem

## Notions of Work and Kinetic Energy

“The work-energy 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: v2 – u2 = 2 a.d (where u - initial velocity, v - final velocity, a -acceleration, d - displacement)
Now multiplying the equation by m/2 we have,

½ mv2 – ½ mu2 = ma.d

½ mv2 – ½ mu2 = F.d (Since ma = F)

### The Work-Energy 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 (xi) to final position (xf), we have
∫_(K_i)^(K_f) dK=∫_(x_i)^(x_f)F dx
where, Kt and Kf are the initial and final kinetic energies corresponding to xi and xf,
or K_f-K_i=∫_(x_i)^(x_f) F dx
From earlier proved equation, it follows that
K_f-K_i =W
Thus, the W.E. theorem is proved for a variable force.

Regarding the work-energy theorem it is worth noting that
(i) If Wnet is positive, then Kf – Ki = positive, i.e., Kf > Ki or kinetic energy will increase and vice-versa.
(ii) This theorem can be applied to non-inertial frames also. In a non-inertial frame it can be written as:

Work done by all the forces (including the Pseudo force) = change in kinetic energy in non-inertial frame.

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