Some good explanation of the basic kinematics. The treatment using position vectors is particularly helpful . The point of interest is related to 2 observers. The relative position of one observer is given in the other observers reference frame, The relative observer is moving. Because he is moving at a constant velocity a fixed vector is related to a pair of constantly changing vectors.
Now by diferentiating the constant vector with respect to time the professor obtains the relationship between the 3 moving vectors. What he has not pointed out or considered is the frame of reference in which the formerly fixed reference vector is moving. The possibilities are:
1 in the existing reference frames
2 in a new unmentioned reference frame.
If in the existing reference frames there is a contradiction. The point p is assumed fixed. Thus one of the vectors cannot move, we need a new reference frame.
Suppose we allow the point to move then it moves in both reference frames. Thus the derivative of the fixed vector cannot be the derivative of the observers relative position plus the relative constant motion. There is an additional motion to be factored in
This last link is a thorough and historically researched refutation of the Lorentz and einsteinium approaches.
The consequence of this observation is discussed in the autodynamics forum. Apparently it affects the notion of a Lorentz transformation.
misconception in this explanation is equating dx/dt with dx'/dt
The following is indeed an elementary point, but since some students have felt puzzled with the expression "Galilean invariance", I should like to add this note.
The prevailing attitude is arrogant
This explanation is confusing, probably due to translation. There are everal ways to measure speed and position, but from the outset all are relative, What is usually left out of any exposition is the subjective gnomon, the point of view of the "uber" observer, you and i. We are usually invited to take the omnipotent position, the all knowing position. We can then verify both observers whom we pick..
There is some concern about p whether it can be observed or not. This may be more relevant to signal transmission that vision ,
The set up is where the premises go awry, We either gave 2 people in cars looking at the Eiffel tower while driving at different speeds or one person sitting on a bench while another drives by looking at the Eiffel tower, in the first case we have 3 frames of reference in the second 2. In both cases one frame has the eiffel tower fixed.
Now the example at the beginning of the post has 3 frames of reference and possibly 4. A fixed observer, a moving train, a moving ball and another ball able to move but fixed currently in the trains reference frame,
We can set up common markers for length measure and time measure. The person on board the train represents a 5th reference frame, and he can communicate with the observer who is fixed.
Both observers measure the speed of the train using the common marker and common time.
Assumption !: they measure the same speed and velocity relative to the fixed observer. He reports the speed to the fixed observer.
Now the observer on the train decides to roll a ball at the stationary ball on the train. He measures the speed of the ball and reports it to the fixed observer, who is also measuring the interaction.
Assumption 2 the measurements are related in some way that is consistent.
The fixed observer measures the speed of the ball as x, and he knows that it is made up of 2 parts: the speed of the train and the speed of the ball relative to the train. He knows this because he measured the ball on the last run when it ws stationary relative to the train and it had the same speed as the train.
He cannot measure the speed of the ball relative to the train, but he can measure the speeds of the train and the ball. By subtraction he can calculate a speed for the ball relative to the train. The observer on the train can measure the relative speed of the ball directly. The above assumption can be verified or falsified.
The whole system is based on measuring relative to the correct reference frame and the 2 assumptions and observations. The observations are in the same units, that is speeds, or velocities.
The standard derivation of the Galilean transformation mixes distance and velocity, thus it mixes which reference frames re moving with which ones are still. It also confuses collinearity with the equality of motion, just because the equality of distance holds . Thus 2 collinear distances are equal but one is in motion the other is not. Tf the motionless one starts to move in the same direction as the moving one its speed or velocity is not coupled to or connected to the one in motion, just as a stationary car can pull out into a stream of traffic when the moving car passes, but then it can accelerate and pass the car in front.
Zeno had a problem: the tortoise and the hair. by all accounts the hare never passes the tortoise, and never will. The observer is duped into lock stepping the tortoise and the hare when in fact they are not coupled or linked. When they are coupled then we can use a Galilean transformation. To be linked the linked motion has to occur entirely within a single reference frame. If this is not the case, ie an object is moving in 2 or more reference frames the independent frames are not coupled and the reference frame that is most appropriate can be chosen.
Why do we want to be able to give an account of an event in alternative reference frames? The main use is not to find the absolute stationary point, but to be able to communicate between observers in different vehicles/frames and confirm the same observations. For any vehicles traveling in a fleet the Galilean transformation enables the members of the fleet to give a common position by sextant and navigation sightings. However as soon as one of the vehicles decouples the Galilean transformation is useless.
the Doppler effect and the Lorentz effect need to be examined together, not as some derived law from the Galilean transformation.
This last link contains a historically researched refutation of the Lorentz and Einsteinian position.