Formula Structure of Energy


EN = KM = Potential Energy of a System


The potential energy of an object times all of the mass not of the object within the closed system containing the object equals all of the kinetic energy of the closed system not of the object times the mass of the object. Potential energy, kinetic energy, and mass are interchangeable.


Formulated by Adam Pellegrino


Note: Currently rewriting definitions concercing N and n … was defining them as accumulations of mass rather than the dispersions of mass: spreadsheet formulas remain the same. 0.5 in interval formulas assumes perfect linear interval distributions, 0.5 is therefore defined as the variable average for the interval.

Any help on translating formulas is appreciated as it is significantly easier to create a functional spreadsheet than it is to adequately express its formulas into words.


All of the following formulas are compiled from a spreadsheet (if you would like to verify the equations it is recommended you use the accompanying spreadsheet to observe the inter-relations):

A = Acceleration = current speed - previous speed = 2(S - d/t) = 2(St - d)/t;

a = Total Average Acceleration = S/T = Ss/D = (0.5A + d/t)/T;

D = Total Distance = sT = Ss/a = s(0.5A + d/t)/a = D0 + sum of all (d) since T0 = sum of all (d).

d = Interval Distance = current total distance - previous total distance = t(interval average speed) = t(S - 0.5A) = 0.5t(2S - A);

S = Speed = aT = Da/s = 0.5A + d/t = S0 + sum of all (A) since T0 = sum of all (A);

s = Average Speed = D/T = Da/S = Da/(0.5A + d/t);

T = Total Time = D/s = S/a = (0.5A + d/t)/a = T0 + sum of all (t) since T0 = sum of all (t);

t = Interval Time = current total time - previous total time = d/interval average speed = d/(S - 0.5A) = 2d/(2S - A);

A/t = Average Rate of Interval Acceleration = 0.5[S2 - (SX-1)2]/d = 0.5(S + SX-1)(S - SX-1)/d;

2Ad/t = A(2S - A) = S2 - (SX-1)2 = (S + SX-1)(S - SX-1);

Da = Ss;

SX-1 = previous S at 1 interval back = S - A;

D0 = Total Distance at T0 of Time Series = D - sum of all (d) since T0 = sum of all (d) up to and including T0;

S0 = Speed at T0 of Time Series = S - sum of all (A) since T0 = sum of all (A) up to and including T0;

T0 = Point on Time Series where Time equals Zero = total time - sum of all (t) since T0 = sum of all (t) up to and including T0;

S2 = 2 sum of all (Ad/t) = 2 sum of all (Ad/t) since T0 + (S0)2 = (SX-1)2 + 2Ad/t;

(S0)2 = S2 - 2 sum of all (Ad/t) since T0 = (S - A)2 - 2[sum of all (Ad/t) since T0 - Ad/t] = (S - A)2 - 2 sum of all (Ad/t) since T0 + A(2S - A) = (S - At)2 - 2 sum of all (Ad/t) since T0 + At(2S - At);

0.5S2 = Total (Ad/t) = sum of all (Ad/t) = sum of all (Ad/t) since T0 + 0.5(S0)2;

Interval Average Acceleration = d/t = S - 0.5A = 0.5(2S - A).


The above formulas are modifiable into acceleratory equations by setting D0, S0, and T0 = 0 at T0. Unless starting at the origin of time itself, D0, S0, and T0 are artificial starting points upon a newly created time series (if you were in a race, you would measure how fast you ran with T0 being the moment the stating gun fired, and by setting the moment of which the gun went off equal to zero, where your starting distance would also be zero, even though time existed before the gun went off, you were on a planet that was already moving before the gun sounded, and your position to the rest of the universe subsequently changed in unison to the motion of the planet). Acceleratory equations express changes in speed caused from acceleration and are only representative of actual speed when the initial at rest speed is correctly factored in. The word “acceleratory” is used here to mean any portion of an equation that changes in value due to the change of another variable relative to a series containing a predefined point of origin, such as T0. T0 automatically includes gravity, and other forces such as electromagnetic and friction into acceleratory equations relative to corresponding non-acceleratory equations. A variable expressed with a subscripted A is acceleratory; variables A and T are not likewise notated as their values are the same as the already presented A and t formulas.

A = Acceleratory Acceleration = current acceleratory speed - previous acceleratory speed = 2(SA - dA/t) = 2(SAt - dA)/t;

aA = Total Average Acceleratory Acceleration = (sum of all A since TA,0)/TA = SA/TA = SAsA/DA = (0.5A + dA/t)/TA;

DA = Acceleratory Total Distance = sATA = SAsA/aA = sA(0.5A + dA/t)/aA = sum of all (dA) since TA,0;

dA = Acceleratory Interval Distance = current acceleratory total distance - previous acceleratory total distance = t(interval average acceleratory speed) = t(SA - 0.5A) = 0.5t(2SA - A);

SA = Acceleratory Speed = aATA = DAaA/sA = 0.5A + dA/t = sum of all (A) since TA,0;

sA = Average Acceleratory Speed = DA/TA = DAaA/SA = DAaA/(0.5A + dA/t);

TA = Total Acceleratory Time = DA/sA = SA/aA = (0.5A + dA/t)/aA = sum of all (t) since TA,0;

t = Acceleratory Interval Time = current total acceleratory time - previous total acceleratory time = dA/interval average acceleratory speed = dA/(SA - 0.5A) = 2dA/(2SA - A);

A/t = Average Rate of Interval Acceleratory Acceleration = 0.5[(SA)2 - (SA,X-1)2]/dA = 0.5(SA + SA,X-1)(SA - SA,X-1)/dA;

2AdA/t = A(2SA - A) = (SA)2 - (SA,X-1)2 = (SA + SA,X-1)(SA - SA,X-1);

DAaA = SAsA;

SA,X-1 = previous SA at 1 interval back = SA - A;

DA,0 = Total Acceleratory Distance at TA,0 of Time Series = DA - sum of all (dA) since TA,0 = 0;

SA,0 = Acceleratory Speed at TA,0 of Time Series = SA - sum of all (A) since TA,0 = 0;

TA,0 = Starting Time of Acceleratory Time Series = TA - sum of all (t) since TA,0 = 0;

(SA)2 = 2 sum of all (AdA/t) since TA,0 = (SA,X-1)2 + 2AdA/t;

0.5(SA)2 = Total(AdA/t) = sum of all (AdA/t) since TA,0;

Interval Average Acceleratory Acceleration = dA/t = SA - 0.5A = 0.5(2SA - A).


Subsequent mixed equations (equations using both acceleratory and non-acceleratory variables) are:

a = SsA/DA = sA(SA + S0)/DA = s(SA + S0)/D;

aA = SAs/D = (S - S0)/T;

d = dA + S0t = dA + t(S - SA);

dA = d - S0t = d - t(S - SA);

D = DA + S0TA + D0 = SAs/aA = s(SA + S0)/a;

DA = D - S0TA - D0 = SAs/aA - S0TA - D0;

D0 = D - S0TA - DA = SAs/aA - S0TA - DA;

DsA = DAs;

DaA = SAs;

DAa = SsA;

s = Da/S = Da/(SA + S0) = DaA/SA = DaA/(S - S0);

sA = DAa/S = DAa/(SA + S0);

S = SA + S0 = aAT + S0 = a(S0TA + DA + D0)/s;

SA = S - S0 = aAD/s = S - (d - dA)/t;

S0 = S - SA = S - aAD/s = (d - dA)/t = (D - DA - D0)/TA;

t = (d - dA)/S0 = (d - dA)/(S - SA);

T = TA + T0 = (S - S0)/aA = (D - DA - D0)/S0 + T0;

TA = T - T0 = (S - S0)/aA - T0 = (D - DA - D0)/S0;

T0 = TA,0 = T - TA = (S - S0)/aA - TA = T - (D - DA - D0)/S0;

S2 = (SA + S0)2 = (SA)2 + 2SAS0 + (S0)2 = (SA)2 + 2SS0 - (S0)2;

(SA)2 = (S - S0)2 = S2 - 2SAS0 - (S0)2 = S2 - 2SS0 + (S0)2;

(S0)2 = (S - SA)2 = (SA - S)2 = S2 - 2SSA + (SA)2 = S2 - 2SAS0 - (SA)2 = (SA)2 + 2SS0 - S2.


More equations manifest when ∆ is used to represent changes in variables:

∆ = Change in;

A = ∆S = ∆SA;

∆(d/t) = ∆(dA/t) = 0.5(A + A X-1);

∆(Ad/t) = Ad/t - (Ad/t)X-1 = (∆A)d/t + (A - ∆A)∆(d/t) = A∆(d/t) + ∆A(d/t - ∆(d/t));

∆(AdA/t) = AdA/t - (AdA/t)X-1 = (∆A)dA/t + (A - ∆A)∆(dA/t) = A∆(dA/t) + ∆A(dA/t - ∆(dA/t);

Ad/t = sum of all ∆(Ad/t);

AdA/t = sum of all ∆(AdA/t);

(Ad/t)X-1 = Ad/t - ∆(Ad/t) = (A - ∆A)(d/t - ∆(d/t);

(AdA/t)X-1 = AdA/t - ∆(AdA/t) = (A - ∆A)(dA/t - ∆(dA/t).


All formulas presented so far can be used for the formalization of mass by letting S = mass, A = change in mass/t, and D = total dispersion of mass. Equations for mass result as follow:

U = Interval Dispersed Mass = current mass - previous mass = 2(M - n/t) = 2(Mt - n)/t;

u = Average Mass = M/T = Mm/N = (0.5U + n/t)/T;

N = Total Dispersed Mass = mT = Mm/u = m(0.5U + n/t)/u = N0 + sum of all (n) = t(average interval mass) + N0 = sum of all (n);

n = Average Interval Summed Mass = current total dispersed mass - previous total dispersed mass = t(averarage interval mass) = t(M - 0.5U) = 0.5t(2M - U);

average interval mass = n/t = (M - 0.5U) = 0.5(2M – U);

M = Object Mass = uT = Nu/m = 0.5U + n/t = M0 + sum of all (U) since T0 = sum of all (U);

m = Average Dispersed Mass = N/T = Nu/M = Nu/(0.5U + n/t);

T = Total Time = N/m = M/u = (0.5U + n/t)/u = T0 + sum of all (t) since T0 = sum of all (t);

t = Interval Time = current total time - previous total time = n/(average interval mass) = n/(M - 0.5U) = 2n/(2M - U);

U/t = Average Interval Dispersion Rate = 0.5[M2 - (MX-1)2]/n = 0.5(M + MX-1)(M - MX-1)/n;

2Un/t = U(2M - U) = M2 - (MX-1)2 = (M + MX-1)(M - MX-1);

Nu = Mm;

MX-1 = previous M at 1 interval back = M - U;

N0 = Total Dispersed Mass at T0 of Time Series = total mass outside of an object at T0 = N - sum of all (n) since T0 = sum of all (n) up to and including T0;

M0 = Object Mass at T0 of Time Series = M - sum of all (U) since T0 = sum of all (U) up to and including T0;

T0 = Point on Time Series where Time equals Zero = total time - sum of all (t) since T0 = sum of all (t) up to and including T0;

M2 = 2 sum of all (Un/t) = 2 sum of all (Un/t) since T0 + (M0)2 = (MX-1)2 + 2Un/t;

(M0)2 = M2 - 2 sum of all (Un/t) since T0 = (M - U)2 - 2[sum of all (Un/t) since T0 - Un/t] = (M - U)2 - 2 sum of all (Un/t) since T0 + U(2M - U) = (M - Ut)2 - 2 sum of all (Un/t) since T0 + Ut(2M - Ut);

0.5M2 = Total (Un/t) = sum of all (Un/t) = sum of all (Un/t) since T0 + 0.5(M0)2.


The above formulas can be modified to become accumulatory equations by setting N0, M0, and T0 at T0 = 0. Mass is seldom looked at from a starting accumulatory standpoint set equal to zero, rather mass is largely viewed with a starting value equal to the weight, size, or composition of an object; historically, speed and mass are valued differently when placed into the same equational structures (starting speed = 0, starting mass = weight), though for no reason other than how we observe. The word accumulatory isn’t found in dictionaries, but intuitively means “of an accumulative nature,” and is used here to mean any portion of an equation that changes in value due to the change of another variable relative to a series containing a predefined point of origin, such as T0. T0 automatically includes gravity, and other forces such as electromagnetic and friction into accumulatory equations relative to corresponding non-accumulatory equations. Accumulatory equations express changes in mass caused from accumulation and are only representative of actual mass when the initial at rest mass is correctly factored in. A variable expressed with a subscripted U is accumulatory; variables of U and t are not likewise notated as their values are the same as the already presented U and t formulas; Accumulatory T is subscripted with an A, as TU is just another title for TA.

U = Accumulatory Interval Dispersed Mass = current accumulatory dispersed mass - previous accumulatory dispersed mass = 2(MU - nU/t) = 2(MUt - nU)/t;

uU = Total Average Accumulatory Dispersed Mass = (sum of all U since TA,0)/TA = MU/TA = MUmU/NU = (0.5U + nU/t)/TA;

NU = Accumulatory Total Dispersed Mass = mUTA = MUmU/uU = mU(0.5U + nU/t)/uU = sum of all (nU) since TA,0;

nU = Accumulatory Interval Dispersed Mass = current accumulatory dispersed mass - previous accumulative dispersed mass = t(interval average accumulatory dispersed mass) = t(MU - 0.5U) = 0.5t(2MU - U);

MU = Accumulatory Mass = uUTA = NUuU/mU = 0.5U + nU/t = sum of all (U) since TA,0;

mU = Average Accumulatory Mass = NU/TA = NUuU/MU = NUuU/(0.5U + nU/t);

TA = Total Accumulatory Time = NU/mU = MU/uU = (0.5U + nU/t)uU = sum of all (t) since TA,0;

t = Accumulatory Interval Time = current total accumulatory time - previous total accumulatory time = nU/interval average accumulatory mass = nU/(MU - 0.5U) = 2nU/(2MU - U);

U/t = Average Rate of Interval Accumulatory Dispersion = 0.5[(MU)2 - (MU,X-1)2]/nU = 0.5(MU + MU,X-1)(MU - MU,X-1)/nU;

2UnU/t = U(2MU - U) = (MU)2 - (MU,X-1)2 = (MU + MU,X-1)(MU - MU,X-1);

NUuU = MUmU;

MU,X-1 = previous MU at 1 interval back = MU - U;

NU,0 = Total Accumulatory Dispersed Mass at TA,0 of Time Series = NU - sum of all (nU) since TA,0 = 0;

MU,0 = Accumulatory Mass at TA,0 of Time Series = MU - sum of all (U) since TA,0 = 0;

TA,0 = Starting Time of Accumulatory Time Series = TA - sum of all (t) since TA,0 = 0;

(MU)2 = 2 sum of all (UnU/t) since TA,0 = (MU,X-1)2 + 2UnU/t;

0.5(MU)2 = Total (UnU/t) = sum of all (UnU/t) since TA,0;

Interval Average Accumulatory Dispersed Mass = nU/t = MU - 0.5U = 0.5(2MU - U).


Subsequent mixed equations (equations using both accumulatory and non-accumulatory variables) are:

u = MmU/NU = mU(MU + M0)/NU = m(MU + M0)/N;

uU = MUm/N = (M - M0)/T;

n = nU + M0t = nU + t(M - MU);

nU = n - M0t = n - t(M - MU);

N = NU + M0TA + N0 = MUm/uU = m(MU +M0)/u;

NU = N - M0TA - N0 = MUm/uU - M0TA - N0;

N0 = N - M0TA - NU = MUm/uU - M0TA - NU;

NmU = NUm;

NuU = MUm;

NUu = MmU;

m = Nu/M = Nu/(MU + M0) = NuU/MU = NuU/(M - M0);

mU = NUu/M = NUu/(MU + M0);

M = MU + M0 = uUT + M0 = u(M0TA + NU + N0)/m;

MU = M - M0 = uUN/m = M - (n - nU)/t;

M0 = M - MU = M - uUN/m = (n - nU)/t = (N - NU - N0)/TA;

t = (n - nU)/M0 = (n - nU)/(M - MU);

T = TA + T0 = TU + T0 = (M - M0)/uU = (N - NU - N0)/M0 + T0;

TA = TU = T - T0 = (M - M0)/uU - T0 = (N - NU - N0)/M0;

T0 = TA,0 = TU,0 = T - TA = T - TU = (M - M0)/uU - TA = T - (N - NU - N0)/M0;

M2 = (MU + M0)2 = (MU)2 + 2MUM0 + (M0)2 = (MU)2 + 2MM0 - (M0)2;

(MU)2 = (M - M0)2 = M2 - 2MUM0 - (M0)2 = M2 - 2MM0 + (M0)2;

(M0)2 = (M - MU)2 = (MU - M)2 = M2 - 2MMU + (MU)2 = M2 - 2MUM0 - (MU)2 = (MU)2 + 2MM0 - M2.


More equations manifest when ∆ is used to represent changes in variables:

U = ∆M = ∆MU;

∆(n/t) = ∆(nU/t) = 0.5(U + U X-1);

∆(Un/t) = Un/t - (Un/t)X-1 = (∆U)n/t + (U - ∆U)∆(n/t) = U∆(n/t) + ∆U(n/t - ∆(n/t));

∆(UnU/t) = UnU/t - (UnU/t)X-1 = (∆U)nU/t + (U - ∆U)∆(nU/t) = U∆(nU/t) + ∆U(nU/t - ∆(nU/t);

Un/t = sum of all ∆(Un/t);

UnU/t = sum of all ∆(UnU/t);

(Un/t)X-1 = Un/t - ∆(Un/t) = (U - ∆U)(n/t - ∆(n/t);

(UnU/t)X-1 = UnU/t - ∆(UnU/t) = (U - ∆U)(nU/t - ∆(nU/t).


Mass can be expressed with speed, acceleration, and distance, without utilizing variables U and N, as follows:

∆M = M - MX-1;

M = MX-1 + ∆M = M0 + sum of all ∆M since T0 = sum of all ∆M since moment before origin of M;

MX-1 = M - ∆M = M0 + sum of all ∆M since T0 - ∆M;

M0 = M at T0 of time series = sum of all ∆M since moment before origin of M up to and including T0;

MOrigin = the value of M upon initial instance of origin, creation;

MAd/t = total ∆(MAd/t) = sum of all ∆(MAd/t) = (MAd/t)X-1 + ∆(MAd/t) = total (MAd/t) - total (MAd/t)X-1;

Total (MAd/t) = sum of all (MAd/t) = total (MAd/t)X-1 + MAd/t = MUAd/t + 0.5M0[S2 - (S0)2] - sum of all [(∆M - T∆M)(Ad/t - ∆(Ad/t))] since T0 = MAd/t + 0.5M0[S2 - 2Ad/t - (S0)2] - sum of all [(∆M - total ∆M)((Ad/t - ∆(Ad/t))] since T0 = Mad + 0.5M0[S2 - 2Ad - (S0)2] - (t - 1)MUAd/t - sum of all [(∆M - total ∆M)((Ad/t - ∆(Ad/t))] since T0;

(MAd/t)X-1 = total (MAd/t) - ∆(MAd/t) = (M - ∆M)(A - ∆A)[d/t - ∆(d/t)] = (M - ∆M)[Ad/t - ∆(Ad/t)] = (A - ∆A)[Md/t - ∆(Md/t)] = [d/t - ∆(d/t)][MA - ∆(MA)];

∆(MAd/t) = MAd/t - (MAd/t)X-1 = ∆M∆A∆(d/t) + [∆MAd/t - M∆A∆(d/t)] + [M∆Ad/t - ∆MA∆(d/t)] + [MA∆(d/t) - ∆M∆Ad/t] = MA∆(d/t) + ∆(MA)[d/t - ∆(d/t)] = M∆(Ad/t) + ∆;M[Ad/t - ∆(AD)] = M∆Ad/t + ∆(Md/t)(A - ∆A);

[∆(MAd/t)]X-1 = (MAd/t)X-1 - (MAd/t)X-2 = [∆M∆A∆(d/t)]X-1 + [∆MAd/t - M∆A∆(d/t)]X-1 + [M∆Ad/t - ∆MA∆(d/t)]X-1 + [MA∆(d/t) - ∆M∆Ad/t]X-1 = [MA∆(d/t)]X-1 + [∆(MA)(d/t - ∆(d/t))]X-1 = [M∆(Ad/t)]X-1 + [∆M[Ad/t - ∆(AD)]]X-1 = (M∆Ad/t)X-1 + [∆(Md/t)(A - ∆A)]X-1.


The term “(∆M - total ∆M)(Ad/t - ∆(Ad/t))” from the total (MAd/t) equations can be written as (total ∆M - ∆M)(∆(Ad/t) - Ad/t), -(∆M - total ∆M)(∆(Ad/t) - Ad/t), or -(total ∆M - ∆M)(AAd/t - ∆(Ad/t)), and still be correct. This relationship establishes what I like to call the Law of Vectors, which states that given numerical sequences of two variables (A and B) the following relationships are true: (A - sum of all A)(B - sum of all B) = (sum of all A - A)(sum of all B - B) = -(A - sum of all A)(sum of all B - B) = -(sum of all A - A)(B - sum of all B).


Energy equations share the same equational structure presented for speed and mass. For the following equations, E = potential energy, Y = rate of kinetic energy, and K = summation of kinetic energy over time.

Y = Kinetic Energy = current potential energy - previous potential energy = 2(E - k/t) = 2(Et - k)/t;

y = Total Average Kinetic Energy = E/T = Ee/K = (0.5Y + k/t)/T;

K = Total Kinetic Energy = eT = Ee/y = e(0.5Y + k/t)/y = K0 + sum of all (k) since T0 = sum of all (k);

k = Interval Kinetic Energy = current total kinetic energy - previous total kinetic energy = t(interval average potential energy) = t(E - 0.5Y) = 0.5t(2E - Y);

E = Potential Energy = yT = Ky/e = 0.5Y + k/t = E0 + sum of all (Y) since T0 = sum of all (Y);

e = Average Potential Energy = K/T = Ky/E = Ky/(0.5Y + k/t);

T = Total Time = K/e = E/y = (0.5Y + k/t)/y = T0 + sum of all (t) since T0 = sum of all (t);

t = Interval Time = current total time - previous total time = y/interval average potential energy = k/(E - 0.5Y) = 2k/(2E - Y);

Y/t = Average Rate of Interval Kinetic Energy = 0.5[E2 - (EX-1)2]/k = 0.5(E + EX-1)(E - EX-1)/k;

2Yk/t = Y(2E - Y) = E2 - (EX-1)2 = (E + EX-1)(E - EX-1);

Ky = Ee;

EX-1 = previous E at 1 interval back = E - Y;

K0 = Total Kinetic Energy at T0 of Time Series = K - sum of all (k) since T0 = sum of all (k) up to and including T0;

E0 = Potential Energy at T0 of Time Series = E - sum of all (Y) since T0 = sum of all (Y) up to and including T0;

T0 = Point on Time Series where Time equals Zero = total time - sum of all (t) since T0 = sum of all (t) up to and including T0;

E2 = 2 sum of all (Yk/t) = 2 sum of all (Yk/t) since T0 + (E0)2 = (EX-1)2 + 2Yk/t;

(E0)2 = E2 - 2 sum of all (Yk/t) since T0 = (E - Y)2 - 2[sum of all (Yk/t) since T0 - Yk/t] = (E - Y)2 - 2 sum of all (Yk/t) since T0 + Y(2E - Y) = (E - Yt)2 - 2 sum of all (Yk/t) since T0 + Yt(2E - Yt);

0.5E2 = Total (Yk/t) = sum of all (Yk/t) = sum of all (Yk/t) since T0 + 0.5(E0)2;

Interval Average Potential Energy = k/t = E - 0.5Y = 0.5(2E - Y).


The above formulas can be modified to become kinematical equations by setting K0, E0, and T0 at T0 = 0. Kinematical equations express changes in potential energy arising from kinetic energy and are representative of actual potential energy when the initial at rest potential energy is properly factored in. The word “kinematical” is termed here to mean any portion of an equation that changes in value due to the change of another variable relative to a series containing a predefined point of origin, such as T0. T0 automatically includes gravity, and other forces such as electromagnetic and friction in kinematical equations relative to corresponding non-kinematical equations. A variable expressed with a subscripted K is kinematical; variables of Y and t are not likewise notated as their values are the same as the already presented Y and t formulas; Kinematical T is subscripted with an A, as TK is just another title (TA = TU = TK) for TA.

Y = Kinematical Kinetic Energy = current kinematical potential energy - previous kinematical potential energy = 2(EK - kK/t) = 2(EKt - kK)/t;

yK = Total Average Kinematical Kinetic Energy = (sum of all Y since TA,0)/TA = EK/TA = EKeK/KK = (0.5Y + kK/t)/TA;

KK = Kinematical Total Kinetic Energy = eKTA = EKeK/yK = eK(0.5Y + kK/t)/yK = sum of all (kK) since TA,0;

kK = Kinematical Interval Kinetic Energy = current kinematical total kinetic energy - previous kinematical total kinetic energy = t(interval average kinematical potential energy) = t(EK - 0.5Y) = 0.5t(2EK - Y);

EK = Kinematical Potential Energy = yKTA = KKyK/eK = 0.5Y + kK/t = sum of all (Y) since TA,0;

eK = Average Kinematical Potential Energy = KK/TA = KKyK/EK = KKyK/(0.5Y + kK/t);

TA = Total Kinematical Time = KK/eK = EK/yK = (0.5Y + kK/t)/yK = sum of all (t) since TA,0;

t = Kinematical Interval Time = current total kinematical time - previous total kinematical time = kK/interval average kinematical potential energy = kK/(EK - 0.5Y) = 2kK/(EK - Y);

Y/t = Average Rate of Interval Kinematical Kinetic Energy = 0.5[(EK)2 - (EK,X-1)2]/kK = 0.5(EK + EK,X-1)(EK - EK,X-1)/kK;

2YkK/t = Y(2EK - Y) = (EK)2 - (EK,X-1)2 = (EK + EK,X-1)(EK - EK,X-1);

KKyK = EKeK;

EK,X-1 = previous EK at 1 interval back = EK - Y;

KK,0 = Total Kinetic Energy at TA,0 of Time Series = KK - sum of all (kK) since TA,0 = 0;

EK,0 = Potential Energy at TA,0 of Time Series = EK - sum of all (Y) since TA,0 = 0;

TA,0 = Starting Time of Kinematical Time Series = TA - sum of all (t) since TA,0 = 0;

(EK)2 = 2 sum of all (YkK/t) since TA,0 = (EK,X-1)2 + 2YkK/t;

0.5(EK)2 = Total(YkK/t) = sum of all (YkK/t) since TA,0;

Interval Average Kinematical Potential Energy = kK/t = EK - 0.5Y = 0.5(2EK - Y).


Subsequent mixed equations (equations using both kinematical and non-kinematical variables) are:

y = EeK/KK = eK(EK + E0)/KK = e(EK + E0)/K;

yK = EKe/K = (E - E0)/T;

k = kK + E0t = kK + t(E - EK);

kK = k - E0t = k - t(E - EK);

K = KK + E0TA + K0 = EKe/yK = e(EK + E0)/y;

KK = K - E0TA - K0 = EKe/yK - E0TA - K0;

K0 = K - E0TA - KK = EKe/yK - E0TA - KK;

KeK = KKe;

KyK = EKe;

KKy = EeK;

e = Ky/E = Ky/(EK + E0) = KyK/EK = KyK/(E - E0);

eK = KKy/E = KKy/(EK + E0);

E = EK + E0 = yKT + E0 = y(E0TA + KK + K0)/e;

EK = E - E0 = yKK/e = E - (k - kK)/t;

E0 = E - EK = E - yKK/e = (k - kK)/t = (K - KK - K0)/TA;

t = (k - kK)/E0 = (k - kK)/(E - EK);

T = TA + T0 = TU + T0 = TK + T0 = (E - E0)/yK = (K - KK - K0)/E0 + T0;

TA = TU = TK = T - T0 = (E - E0)/yK - T0 = (K - KK - K0)/E0;

T0 = TA,0 = TU,0 = TK,0 = T - TA = T - TU = T - TK = (E - E0)/yK - TA = T - (K - KK - K0)/E0;

E2 = (EK + E0)2 = (EK)2 + 2EKE0 + (E0)2 = (EK)2 + 2EE0 - (E0)2;

(EK)2 = (E - E0)2 = E2 - 2EKE0 - (E0)2 = E2 - 2EE0 + (E0)2;

(E0)2 = (E - EK)2 = (EK - E)2 = E2 - 2EEK + (EK)2 = E2 - 2EKE0 - (EK)2 = (EK)2 + 2EE0 - E2.


More equations manifest when ∆ is used to represent changes in variables:

Y = ∆E = ∆EK;

∆(k/t) = ∆(kK/t) = 0.5(Y + Y X-1);

∆(Yk/t) = Yk/t - (Yk/t)X-1 = (∆Y)k/t + (Y - ∆Y)∆(k/t) = Y∆(k/t) + ∆Y(k/t - ∆(k/t));

∆(YkK/t) = YkK/t - (YkK/t)X-1 = (∆Y)kK/t + (Y - ∆Y)∆(kK/t) = Y∆(kK/t) + ∆Y(kK/t - ∆(kK/t);

Yk/t = sum of all ∆(Yk/t);

YkK/t = sum of all ∆(YkK/t);

(Yk/t)X-1 = Yk/t - ∆(Yk/t) = (Y - ∆Y)(k/t - ∆(k/t);

(YkK/t)X-1 = YkK/t - ∆(YkK/t) = (Y - ∆Y)(kK/t - ∆(kK/t).


(S0)2 ties all of the energy equations together:

Y = U(S0)2 = U(S - SA)2 = ∆M(S0)2 = ∆M(S - SA)2;

y = M(S0)2/T = M(S - SA)2/T;

yK = MU(S0)2/T = MU(S - SA)2/T;

K = N(S0)2 = N(S - SA)2;

KK = NU(S0)2 = NU(S - SA)2;

K0 = N0(S0)2 = N0(S - SA)2;

k = n(S0)2 = n(S - SA)2;

kK = nU(S0)2 = nU(S - SA)2;

E = M(S0)2 = M(S - SA)2;

EK = MU(S0)2 = MU(S - SA)2;

E0 = M0(S0)2 = M0(S - SA)2;

e = m(S0)2 = m(S - SA)2;

eK = mU(S0)2 = mU(S - SA)2;

(S0)2 = ∆M/Y = yKT/MU = yT/M = U/Y = K/N = KK/NU = K0/N0 = k/n = kK/nU = E/M = EK/MU = E0/M0 = e/m = eK/mU = (S - SA)2 = (SA - S)2 = S2 - 2SSA + (SA)2 = S2 - 2SAS0 - (SA)2 = (SA)2 + 2SS0 - S2 = S2 - 2 sum of all (Ad/t) since T0 = (S - A)2 - 2[sum of all (Ad/t) since T0 - Ad/t] = (S - A)2 - 2 sum of all (Ad/t) since T0 + A(2S - A) = (S - At)2 - 2 sum of all (Ad/t) since T0 + At(2S - At) = S2 + 2[MUAd/t - total(MAd/t) since T0 - sum of all [(∆M - total ∆M since T0)(Ad/t - ∆(Ad/t))] since T0]/M0 = S2 - 2[total (MAd/t) since T0 - MUAd/t + sum of all [(∆M - total ∆M since T0)(Ad/t - ∆(Ad/t))] since T0]/M0.


To conclude, two additional structurally correct equations are also discovered:

eSystem = (M + dispersed M)(S + S of dispersed M)2, where total mass within a system = (M + dispersed M); and what I term as the “Potential and Kinetic Energy Equality” of EN = KM.


EN = KM = MN(S0)2 = Potential energy of a system

Potential energy of an object = Mass times (S0)2 = M(S0)2

Kinetic energy from an object = Total dispersed mass times (S0)2 = N(S0)2

Both potential and kinetic energy are relative to the mass of an object in question


Adam Pellegrino; Copyright 2008



other links


Formula Structure of Energy Spreadsheet


My Book "Thermal Oblivion" ... might not be perfectly current, or accurate as it is somewhat theoretical


If you are interested in a working copy of the speadsheet, send me a message and let me know the extention you need it as.

Note: I use openoffice (free at openoffice.org) and am limmitted to its save extensions.

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