Deriving kinematics equations for constant jerk, jounce, etc.
Don't google this. Come up with a derivation!
I've decided to challenge myself by rederiving the regular kinematics equations for a constant jerk situation (which might arise in spacecraft rocket
burns, when the mass decreases and the corresponding acceleration increases because the force exerted by the burn remains constant).
So far I have derived the simple equations for the changes in acceleration, velocity, and displacement. These are:
a=a0+jt
a2-a02=2j(v-v0)
v=v0+a0t+1/2jt2
v=v0+at-1/2jt2
v=v0+1/2(a+a0)t
s=s0+v0t+1/2a0t2+1/6jt3
s=s0+vt-1/2a0t2+1/3jt3
But I am trying to find the analog to the extremely useful equation
v2-v02=2a(s-s0)
It has no term for time, which is useful if you are looking for other parameters.
If the cubic analog to the square velocity formula works, it should simplify to the above formula when jerk is set to zero.
I would guess you would have to start by defining time in terms of the quadratic formula or the cubic formula. (The cubic formula is a monster!) Maybe setting them equal to each other would be a good first step (albeit a lengthy one!).
After this I'll attempt to expand into jounce and the domain of quartics. (Fun fact: the derivative of jounce is snap, its derivative is crackle, and
the derivative of crackle is pop!)
[Edited on 25.2.2014 by Brain&Force]
At the end of the day, simulating atoms doesn't beat working with the real things...
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