ABOUT THE v VALUE IN (c+v) (c-v) MATHEMATICS

In (c+v)(c-v) mathematics, the v value is a dynamic variable that can change moment by moment. The speeds, directions of motion, and current positions of reference systems are the components that determine the v value.

In the animations here, you can see in detail how the v value is formed in Doppler Triangles and Doppler Quadrangles. Mathematical equations are not used here. For information on mathematical equations, you can refer to the Doppler Triangle and Doppler Quadrangle pages.

EXPLANATION:
You can use the animation for both Doppler Triangle and Doppler Quadrangle.
You can drag the red points in the animation to any position with the mouse and analyze different scenarios.

In the animation, the reference systems send signals to each other.
The slider simulates the time between the emission and arrival of the signal.
The blue arrow/arrows show the distance traveled by the airplane(s) during the signal arrival time.
When the slider is at its final position (signal arrival time), the animation displays the positions of Image Objects.
When the slider is at its final position, the Black Arrows indicate how the v value is formed for each reference system in the current situation.
As seen in the animation, the Black Arrows are always equal in magnitude, which shows that the v value is a common value for both reference systems.
The text boxes in the bottom right corner show the speeds of the reference systems for the set situation and calculate the v value.
The Black Arrows representing the v value in the animation indicate distance, not speed. x=v.tΔ

Here is a reminder about the v value:
±v value alone shows the speed of approach/recession between two reference systems. Additionally, it represents the amount of deviation from the speed of light.
When written as a component of c, i.e., (c±v), it represents the signal speed. It indicates the speed of the signal sent according to the reference system sending it.

Radio Buttons set the operating mode as either Doppler Triangle or Doppler Quadrangle.
Check Boxes control the visibility of the existing lines.

EXPLANATION:
This animation is a dynamic version of the above animation. After setting the animation to any position, pressing the Play button will cause the airplane(s) to move in the direction of the blue arrow.

I recommend playing the animation after bringing the slider to its final position.

The purpose of this simulation is to highlight the variability of the v value. While the airplane(s) are in motion, you can track the change in the v value from the text box showing the v value in the bottom right corner.

Demo Buttons: Set the animation to a position for easy viewing. Pressing the Play button will start the animation.
Reset Button: Returns the animation to its state at the moment the Start Button was pressed.
Free Button: Allows you to set the animation freely.

Another point I want you to pay attention to is that, as seen in the animation, the Source Object speed and the Image Object speed are not equal. This difference is called Speed Shift. You can find more information about Speed Shift in the Post-Book Publications section.