And the PF community did a great job of discussing some of the physics and mathematics involved.In summary, the conversation centered around the vector equation ##\frac{|V_0 \times V_f|}{g} = R## and its lack of use and teaching in projectile motion. The article "Convenient Equations for Projectile Motion" from the American Journal of Physics was brought up as a useful resource for understanding and utilizing this equation. The AAPT was ultimately very supportive and allowed for the use of their material in the discussion on Physics Forums.
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Introduction
In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our attention the vector equation  ##\frac{|V_0 \times V_f|}{g} = R## and lamented the fact that:
“Equally unused, untaught and apparently not even assigned as a “show that” exercise is Equation (4) that identifies the range as the magnitude of the cross product of the initial and final velocity divided by g.”
In this article, we reproduce and make use of equations presented in a 60s vintage article from the American Journal of Physics entitled “Convenient Equations for Projectile Motion”. The article abstract indicates:
“Quaternion multiplication of the basic vector equations for uniformly...

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Nicely written! Great job!

How hard was it to get the AAPT to approve your use of their material?
 
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Thanks for your kind comment. AAPT (or their representatives) were very helpful - I think they were happy for a long-archived article in AJP to get some renewed attention. Certainly there was something of a process to be followed (as you would expect) but at the end of the day the permission granted was very generous in extent.
 
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1. What are quaternions and how are they used in projectile motion?

Quaternions are a mathematical concept used to represent rotations in three-dimensional space. They are used in projectile motion to calculate the orientation of an object as it moves through space.

2. How do quaternions differ from other methods of representing rotations?

Unlike other methods such as Euler angles or rotation matrices, quaternions do not suffer from gimbal lock, which can cause errors in calculations. They also have a more intuitive geometric interpretation.

3. Can quaternions be used to calculate the trajectory of a projectile?

Yes, quaternions can be used to calculate the trajectory of a projectile. By combining the quaternion representation of the object's orientation with the equations of motion, the position and orientation of the object can be determined at any point in time.

4. Are quaternions necessary for accurate projectile motion simulations?

No, quaternions are not necessary for accurate projectile motion simulations. Other methods such as Euler angles or rotation matrices can also be used. However, quaternions may offer advantages in terms of efficiency and accuracy in certain situations.

5. How do I convert between quaternions and other rotation representations?

Converting between quaternions and other rotation representations can be done using mathematical formulas or conversion functions provided by software libraries. It is important to understand the differences between the representations and their limitations in order to make accurate conversions.

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