study & discuss their (great?) work, energy
WE=m*a*d
using linear_acceleration a (=v/t, =2*((vi/t)+(d/(t*t)), =F/m)
& the force(d) distance d (=va*t)
of a mass m (=F/a).
That (linear accelerated) force is F=m*a,
where the average (accelerated) speed (velocity) va (=(vi+vf)/2),
& speed (velocity, difference) is v=vf-vi,
for final_speed vf,
minus initial_speed vi.
((Even) although it would also be possible
to use factoring (an initial unit_speed of 1 m/s), instead).
The (moving) kinetic_energy is
KE=m*v*va, (pronounced key), or
KE=m*((v^2)/2)+v*vi) using initial_speed vi, or
KE=m*((v^2)/2)-v*vf) with (respect to) final_speed vf.
The potential_energy
PE=m*g*h (pronounced pea, as in pee)
is a mass m
g accelerated (fallen,
multiplied by)
height h=d distance.
PE=Wt*h
is (the force F=) weight Wt (=m*g)
multiplied by height h.
(No distance fallen_able, is no potential_energy.)
Equating (both energies)
PE=KE (pronounced peek)
m*g*h=m*v*va
without mass (divided from both sides) is
g*h=v*va
is only linear_acceleration
a*d=v*va
a*d=(vf-vi)*((vi+vf)/2)
of the distance d,
a*d=((vf^2)-(vi^2))/2, *2
a*d*2=(vf^2)-(vi^2), (pronounced add too).
That's standard mechanics.
(Anything wrong there?)
Swapped sides
(vf^2)-(vi^2)=a*d*2, +(vi^2)
(vf^2)=(vi^2)+a*d*2, ^0.5
vf=((vi^2)+a*d*2)^0.5
is the final_speed (velocity)
(of linear_acceleration).
The speed difference (velocity)
v=vf-vi, vf=((a*d*2+(vi^2))^0.5)
v=((a*d*2+(vi^2))^0.5)-vi
is final_speed (velocity) vf,
minus the initial_speed (velocity) vi.
Momentum('s impulse)
mom=m*v
is mass m
multiplied by (the accelerated) speed difference (velocity) v.
mom=m*(((a*d*2+(vi^2))^0.5)-vi)
mom=m*((a*d*2+(vi^2))^0.5)-m*vi
is the final_momentum momf=m*vf
(pronounced mumf, as in eating fast),
minus the the initial_momentum
momi=m*vi (pronounced mommy).
The average momentum
moma=m*va (pronounced mama)
is the mass m
multiplied by the average (accelerated) speed (velocity) va=(vi+va)/2.
Any questions?
Since KE & mom
only use mass & speeds,
& all energy can be equated to KE,
then it seems imaginable
to equate all energies into impulse(s)
of mom=F*t.
Mom=E/va.
Mother nature (pronounce Eva (the German Eve);
or else Elva
from James Cameron's Avatar).