APPLICATION: RELATIVE COMPARISON INTEGRATION
The multi-combinational mathematics (or theory) has great influenced in almost all areas of sciences. One of its influence area is calculus ( named as “multi-combinational calculus”). The multi-combinational calculus is divided into two main parts. That is relative comparison derivation and relative comparison integration.
The relative comparison integration is a technique I developed for of comparing the initial sequential functions of different variables from their final sequential functions.
APPLYING THE CONCEPT
The speed v0m/s, v1m/s, v2m/s, v3m/s, and v4m/s of bodies 0, 1, 2, 3, and 4 in straight line from a point 0 in the times: t0seconds, t1seconds, t2seconds, t3seconds, and t4seconds respectively is given by the fitted multi-combinational velocity equations,
v0v1v2=24t0t1t2-3t0t1t2 ……(1)
v1v2v3=6t1t2t3+4t1t2t3 ……..(2)
v2v3v4=10t2t3t4+8t2t3t4 ......(3).
Calculate the distance s3 and s4 from 0 at the end of time t0=3s, t1=2s, t2=4s, t3=1s and t4=6s of the bodies 0, 1, 2, 3, and 4 if the initial distance of body 0 and body 1 is s0=108 and s1=144 respectively.
SOLUTION
s0s1s2=∫(v0v1v2)d(t0t1t2)=3(t0t1t2)2-0.375(t0t1t2)2+k1...............(4)
s1s2s3= ∫(v1v2v3)d(t1t2t3)=0.75(t1t2t3)2+0.5(t1t2t3)2+k2….......(5)
s2s3s4= ∫(v2v3v4)d(t2t3t4)=1.25(t2t3t4)2+(t2t3t4)2+k3…………(6)
At the point 0 I.e the starting point t0=0, t1=0, t2=0, t3=0, t4=0 and s0=0, s1=0, s2=0, s3=0, s4=0.Therefore substituting expression t0=0, t1=0, t2=0, t3=0, t4=0 and s0=0, s1=0, s2=0, s3=0, s4=0 into equation(4), (5),and (6) we have k1=0, k2=0, k3=0.
Applying the methods of relative comparison, we have,
s0=[∫(v0v1v2)d(t0t1t2) /∫(v1v2v3)d(t1t2t3)]
s1=s4[∫(v1v2v3)d(t1t2t3)/∫(v2v3v4)d(t2t3t4)]
This implies that;
108=s3[1512/80]
S3=5.71m.
and;
144=[80/1296]
S4=2332.8m.
Hence the distances of the bodies 3 and 4 are 5.71 and 2332.8 respectively.
REFERENCE
D T Whiteside(ed). The Mathematical Papers of Isaac Newton(Volume1). (Cambridge University Press, 1967)
