By the principle of Newtonian relativity, two system of co-ordinates moving uniformly and in a straight line relative to each other are equivalent, i.e., the physical laws which hold good in one system, also hold good in the other. Let system II be moving, relative to system I, uniformly and in a straight line with velocity v. Body A is moving in the same direction with velocity v 1 relative to system I (and therefore with velocity (v 1 – v) relative to system II). A constant force F acts on the body A for a certain period t, in the same line as velocities v and v` and changes the body's velocity relative to system I form v 1 to v 2 . The body's change in kinetic energy will be:
In system I 
, and in system II 
[(v 2 – v) 2 – (v 1 – v) 2 ] = 
(v 2 2 – v 1 2 ) – mv (v 2 – v 1 ),
i.e. less, The change in kinetic energy is therefore different in different system of co-ordinates. How can this be reconciled with Newton's principle of relativity?
Text Solution
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Sol. The principle of relativity demands that the same physical laws should obtain in the two system under consideration, and in particular the law of conservation of energy, according to which the change of energy of a body must equal the work done by external forces. Therefore, in system I the following relationship must obtain
(v 2 2 – v 1 2 ) = Fs ……... (1)1
Where s is the distance traveled by a body in system I in the time during which its velocity rises from v 1 to v 2 .
(v 2 2 – v 1 2 ) – mv (v 2 – v 1 ) = Fs 1 , …...… (2)
Where s 1 is the distance traveled by a body in system II in the same time. But since the velocity of a body in systems I and II is not the same, neither are s and s 1 the same. In fact, since a body moves, under the impulse of a force F, with acceleration F/m, in
system I we have: s = v 1 t + 
. 
,
and in system II correspondingly
s 1 = (v 1 – v) t + 
. 
.
Therefore s–s 1 = vt . But since F/m = a= (v 2 – v 1 )/t,
T = 
.m
and s – s 1 = 
and therefore F (s – s 1 ) = mv (v 2 – v 1 ).
Thus the work of outside forces in system I is as much greater than the work of outside forces in system II as the change in kinetic energy in system I is greater than the change in kinetic energy in system II. And since the change in energy in system I equals the work done by external forces, this also holds for system II. Therefore, Newton's principle of relativity is not broken.
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