Consider two inertial systems x, y and x\' , y\' such that for t = 0 their origi
ID: 1836260 • Letter: C
Question
Consider two inertial systems x, y and x' , y' such that for t = 0 their origins coincide and the dashed system moves with constant velocity V = (v, v) along the bisector of the x, y-plane. In the x, y system, a stone is thrown vertically from the origin with the initial speed v0. Illustrate the invariance (more carefully one should say the covariance) of Newton equations with respect to the Galilean transformations for this example by finiding the equation of the motion in x, y and x', y' systems and by showing that they are related by the corresponding Gallilean transform.
Explanation / Answer
From Gallilean transformation equation ,relation between x ' and x in two inertial sysytems is
x ' = x - vt
Where v = velocity of one inertial system with respect to another. and which is constant
Differentiating with respect to time you get ,
dx'/dt = dx/dt - d(vt)/dt
u ' = u - v -----( 1)
Where u ' and u are the velocities in both inertial systems respectively.
Differentiating equation(1) with respect to time you get ,
(du ' /dt) = (du/dt) - (dv/dt )
a ' = a Since v is constant
Multiplying both sides by m you get,
ma ' = m a
m ' a ' = ma Since m ' = m i.e., in Gallilean transform mass doesnot varies.
F ' = F
Therefore Newton law of motion are invariance.