Add new article: The physics equations of motion for constant position, velocity, acceleration, jerk, snap, crackle, and pop

[ElectricRCAircraftGuy]

I need to write an article on gabrielstaples.com called "The Physics Equations of Motion for Constant x (position, acceleration, jerk, etc.)".

Meticulously derive all of them, easy enough for an average calculus student in high school to understand.

Use MathJax for writing the equations, just like https://math.stackexchange.com/ uses.

See:

1. MathJax basic tutorial and quick reference
2. [my answer] How to draw a stretched vertical bar to indicate the bounds (upper and lower limits) after taking the anti-derivative of a definite integral
3. To try to add MathJax to your GitHub pages website: https://stackoverflow.com/a/72931039/4561887

Solve them with the quadratic formula, as necessary: https://en.wikipedia.org/wiki/Quadratic_formula

Also, use the Python SymPy library to show how to symbolically solve them with Python. See my answer to get started: SymPy symbolic Python library demo: symbolically solving math and integrals & pretty-printing the output

So far, I have personally solved for constant jerk only.

Constant jerk

Constant jerk is helpful because real-life actuators can't instantly change acceleration. Rather, when commanded from one acceleration to another, there is some relatively-constant jerk they can apply to make that change. To model this time the actuator actually takes to change from one acceleration to another, use the constant jerk equation, with the appropriate max jerk inserted as the jerk value to see how long it takes to change from one acceleration to another.

Note also that F = m*a, so a = F/m. It may be helpful to think of it this way as well: acceleration is proportional to Force, and an actuator cannot instantly change the Force. Rather, it can change the Force at some fixed dF/dt max jerk rate. So, again, the constant jerk equations help you see how long it will take to change the Force, and hence the acceleration.

$$\large{v(t) = v = \frac{jt^2}{2} + a_0t + v_0}$$

$$\large{a(t) = j*t + a_0}$$

Constant snap

Someone gave me these; I need to solve for these and verify them myself.

$$\large{j_{k+1} = j_k + snap*dt}$$

$$\large{a_{k+1} = a_k + j_kdt + \frac{1}{2}snap*dt^2}$$

$$\large{v_{k+1} = v_k + a_kdt + \frac{1}{2}j_k*dt^2 + \frac{1}{6}snap\dt^3}$$