Construct a Diagram that Illustrates The Galilean Law of Addition of Velocities

[Math Amateur]

TL;DR Summary

I am reading Bernard Schutz book "A First Course in General Relativity" ... Using mathematics plus text Shutz explains Galileo's Law of Addition of Velocities ... However, Shutz does not provide a diagram illustrating the law... even though this would be most helpful in comprehending the nature of the Law ... ...

I have tried to construct an accurate and pedagogically effective diagram but am quite lost ...

I am reading "A First Course in General Relativity" by Bernard Schutz.

In Chapter1 on Special Relativity Shutz writes:

"... the principle of relativity is not at all a modern concept: it goes back all the way to Galileo's hypothesis that a body in a state of uniform motion remains in that state unless acted upon by some external agency. It is fully embodied in Newton's Second Law, which contains only accelerations, not velocities themselves. Newton's laws are, in fact, all invariant under the replacement

v(t) --> v'(t) = v(t) - V

where V is any constant velocity. This equation says that the velocity v(t) of an object relative to one observer becomes v'(t) when measured by a second observer whose velocity relative to the first is V. This is called the Galilean law of addition of velocities."

Can someone please construct an accurate and pedagogically effective diagram illustrating the law ... I have tried to construct such a diagram but i am quite lost ...

Peter

 

[PeterDonis]

Can someone please construct an accurate and pedagogically effective diagram illustrating the law

First, that's going to be hard to do unless you can be more specific about your requirements. "Accurate and pedagogically effective" is subjective.

Second, why do you need a diagram? The statement that Newton's laws are invariant under the given replacement is a mathematical fact, which is easily checked--in fact Schutz demonstrates it right after the paragraph you quoted. Isn't that sufficient?

 

[Ibix]

Pages 16 and 17.

 

[Orodruin]

Googled ”velocity addition vector”. Lots of hits, including https://pressbooks.online.ucf.edu/phy2053bc/chapter/addition-of-velocities/

 

[Math Amateur]

Thank you for your help, Orodruin, very much appreciate your help ...

Peter

 

[Math Amateur]

Pages 16 and 17.

Thanks ibix ... will study these pages shortly ...

 

[robphy]

In my opinion, for

an accurate and pedagogically effective diagram illustrating the law [Galileo's Law of Addition of Velocities ]

one must first draw its analogue for Euclidean geometry,
in order to later understand what happens in Special Relativity [this subforum]
and in the common-sense familiar Galilean case.

Allow me to setup them up... leaving the details for you [at least for now].

---

In the figure below,
express
the "slope of OS [with respect to OP]" (PS/OP)
in terms of
the "slope of ON [with respect to OP or OM]" (MN/OM)
and
the "slope of OS [with respect to ON]" NS/ON).
Everything can be done using ratios of segments
(and one can use some trigonometric intuition to guide you).

Note: in this Euclidean geometry, NS is Euclidean-perpendicular to ON
since, for radius vector ON, the segment NS is tangent to the Euclidean-circle.

---

Then repeat for this figure in special relativity.
That is,
express
the "velocity of OS [with respect to OP]" (PS/OP)
in terms of
the "velocity of ON [with respect to OP or OM]" (MN/OM)
and
the "velocity of OS [with respect to ON]" NS/ON).
(The method is almost the same....
You might wish to follow your earlier steps and see what has become of them in this case.
However, you'll have to accept that
in this [Minkowski spacetime] geometry, NS is Minkowski-perpendicular to ON
since, for radius vector ON, the segment NS is tangent to the Minkowski-circle.)

---

And, now finally,
repeat for this figure in Galilean relativity.
That is,
express
the "velocity of OS [with respect to OP]" (PS/OP)
in terms of
the "velocity of ON [with respect to OP or OM]" (MN/OM)
and
the "velocity of OS [with respect to ON]" NS/ON).
(The method is almost the same....
You might wish to follow your earlier steps and see what has become of them in this case.
However, you'll have to accept that
in this [Galilean spacetime] geometry, NS is Galilean-perpendicular to ON
since, for radius vector ON, the segment NS is tangent to the Galilean-circle.
)