Question

Derive newton's 2nd law of motion from galilean transformation

Answer 1

Suppose there are two reference frames (systems) designated by S and S' such that the co-ordinate axes are parallel (as in figure 1). In S, we have the co-ordinates $\{x,y,z,t\}$ and in S' we have the co-ordinates $\{x',y',z',t'\}$. S' is moving with respect to S with velocity $v$ (as measured in S) in the $x$ direction. The clocks in both systems were synchronised at time $t=0$ and they run at the same rate.

Answer 2

$\huge \bf{ \red{ANSWER}}$

$\sf{GALILEAN \: TRANSFORMATIONS}$

Consider two frames S and S' of reference one at rest and other is moving with uniform velocity v.

Let O and O' be the observers situated at the origins of S and S' respectively.

They are observing the same event at any point P.

Let two frames be parallel to each other i.e. X'-axis is parallel to X-axis . Y'-axis is parallel to Y-axis, Z'-axis is parallel to Z-axis.

Let the coordinates of P(x,y,z,t) and (x',y',z',t') relative to origins O and O' respectively.

The choice of the origins of two frames is such that their origins concide at time

t=0

and

t'=0

Case 1
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Let the frame S' have the velocity v only in X' direction.

Then O' has velocity v only along X'-axis.

The two systems can be combined to each other by the following equations

(x'=x-vt
y'=y
z'=z
t'=t)...........(1)

Case 2
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Let the frame S' have velocity v along any straight line in any direction such that

v= ivx+jvy+kvz

After time t, the frame S' separated from S by distance tvx,tvy,tvz along x,y,z axes respectively.

then two systems can be related by the following equations.

(x'=x-tvx

y'=y-tvy

z'=z-tvz

t'=t)............(2)

Transformations (1) and (2) are called galilean transformations.

$\huge \pink{THANKS}$