An interesting Note on Acceleration

[Capiert]

Proposal:

If I could bend (=modify)
 Newton’s (3) laws
 ((in)to)
 the way
 I want
 for a consistent continuum,
 I would deal (mostly) with averages

 e.g.
 for (inverse_)time, distance, speed, momentum (& its squared) & mass*Force.

Disclaimer:

(0.
(Some people might think,)

What goes up
 must come down. 

It’s called the 0^th(=zero_th) law
 because he(=Newton) didN’T invent it.
Joke aside.
 ..But (otherwise) here..)

(To be consistent, & thus rebose.)

Preparation:

The (average_)distance d_a
 divided
 by the (average_)time t_a

 is the average_speed v_a=d_a/t_a.

(I have to (repeatedly) say that “average_”
 so often,
 (because (I try))
 NOT to be misunderstood,
 if I were to (simply) say,
 (the average_)
 speed is d/t.)

1.
The average_momentum

 mom_a=m_a*v_a

 is a(n average_)mass m_a
 (coefficient, factor)
 multiplied
 by its average_speed v_a.

2.
To be consistent((ly) based)
 on the same basis, (then)

 the (average_)mass*(average_)Force
 m_a*F_a=mom_a²/d_a

 is (simply?)
 the average_momentum squared
 mom_a²=m_a²*v_a²/d_a

 (then)
 divided
 by the (average (accelerated) traveled) distance d_a.

(That would be
 my substitute
 for mass*Vis_viva

 &/or
 (what seems to me)
 the (ERRORFUL=misguiding=misleeding,
 (interactive)
 work_)energy (concept
 WE=F*d
 which is
 wrongly (proportioned, to only) force F
 multiplied by distance d
 & (thus) needing an extra mass m factor
 as Ewert proposed 1996.)

Or
 the kinetic_energy_(difference, concept)
 KE_d=m*v_d*v_a
 which uses the speed(_difference)
 v_d=v_f-v_i
 for the final_speed v_f
 minus the initial_speed v_i.

 mom_a²/d_a=m_a*(m_a*a_a)
 is the (average_)mass m_a
 multiplied
 by the average_Force F_a= m_a*a_a
 of a(n average_)mass m_a
 multiplied by the average_(linear_)acceleration a_a=v_a²/d_a.

 mom_a²/d_a=m_a*F_a
 which I would love
 to simply call
 mass*Force(_average).

3.
 [Newton’s
 3^rd law
 of] repulsion
 (of((=for)

 the average) mass*Force)
 explains how
 colliding (accelerating) masses
 are equally proportioned.

(mom_a1²/d_a1=m_a1*F_a1)+(mom_a2²/d_a2=m_a2*F_a2)=0.

Again
 (m_a1*F_a1)+(m_a2*F_a2)=0, m_a*F_a=m_a²*a_a
 (m_a1²*a_a1)+(m_a2²*a_a2)=0

 (mom_a1²/d_a1)+(mom_a2²/d_a2)=0.

Note:

2b. (=To_be)

A general
average_acceleration

 a_an=v_aⁿ/d_a^n-1

 could be based
 (up)on
 the “exponential” average_speed

 v_aⁿ= a_an*d_a^n-1

 (for any n>1, e.g. beyond 1)
 which is
 an(y) average_acceleration
 (even NON_linear kinds=types),
 where “a” is just a symbol
 ((only) meaning (a=the) motion;
 NOT necessarily acceleration!).

“n” is just (for) the order of motion
 where
 0 does NOT move?
 1 is the (average_)speed (=NO_acceleration)
 2 is the average_linear_acceleration
 3 is the average_NON linear_acceleration of the 1^st order
 4 is the average_NON linear_acceleration of the 2nd order
 etc.
 

n<1
Exponent
 n values below(=less_than) 1,
 decelerate.

Edited July 12, 2023 by Capiert
Corrections