Define K.E. and derive its relation.
OR
Prove that K.E. = K.E. = $\frac{1}{2} m v^{2}$
Difficulty: Medium
Kinetic Energy:
The energy possessed by a body due to its motion is called its kinetic energy.
Derivation of K.E:
Consider a body of mass m moving with velocity v. The body stops after moving through some distance S due to some opposing force such as the force of friction acting on it. The body possesses kinetic energy and is capable to do work against opposing force F until all of its kinetic energy is used up.
K.E. of the body = Work done by it due to motion
K.E = FS.......... (i)
Vi = v
Vf = 0
As F = ma
A = $-\frac{F}{m}$
Since motion is opposed, hence, a is negative. Using 3rd equation of motion:
$2aS=V_f^2- V_i^2$
$2 (- F/m) S = (0)2 – (v)2$
F S =$\frac{1}{2} m v^{2}$........(ii)
From Eq. (i) and (ii), we get
K.E. = $\frac{1}{2} mv^{2}$.......(iii)
Equation (iii) gives the K.E. possessed by a body of mass m moving with velocity v.
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