Fundamental Jounal of Mathematical Physics Vol. 3 Issue 1 13 Pages 33-44 Published online at http://www.fdint.com/ ESTIMATION MODELS USING MATHEMATICAL CONCEPTS AND NEWTON S LAWS FOR CONIC SECTION TRAJECTORIES ON EARTH S SURFACE Uniesitas Pendidikan Indonesia JI DR Setyabudhi No. 9 Bandung Indonesia e-mail: andikaaisetyawan@yahoo.co.id Abstact Suppose an object is thown obliquely nea the eath s suface. Theoetically its tajectoy is a paabolic. We hae known that the combinations of Unifom Linea Motion (ULM) with constant elocity on X-axis and Unifomly Acceleated Linea Motion (UALM) on Y-axis poduce a paabolic motion. Based on mathematical concepts and Newton s Laws of motion a semicicula motion also can be consideed as the combinations of diffeent motion on X-axis and Y-axis. This theoetical study attempted to compae paabolic motion and semicicula motion. We hae poed that a semicicula motion on eath s suface with small adius is a paticula case of paabolic motion with diffeent coefficients of highest ode at time t. 1. Intoduction This aticle compaes between paabolic motion and semicicula motion. We will also inestigate equations on X-axis and Y-axis using mathematical concepts and Newton s law of motion. Theoetically in physics if we thow an object obliquely nea the eath s suface the tajectoy will be a paabolic. What if the tajectoy fom an object thown obliquely nea the eath s suface is a semicicula? What Keywods and phases: paabolic motion semicicula motionulm UALM. Receied Septembe 6 1 13 Fundamental Reseach and Deelopment Intenational
34 combinations on X-axis and Y-axis will be? This aticle will combine mathematical concepts like paabolic and cicle equations and then compae with the equations which ae deied fom Newton s law of motion. follows:. Theoetical Reiew Befoe we discuss about semicicula motion we ecall the paabolic motion as.1. Paabolic motion Suppose an object is thown obliquely nea the eath s suface with initial elocity and α is an angle between initial elocity ecto and X-axis Figue 1. Paabolic tajectoy. In Physics Figue 1 aboe can be expessed into equations on X-axis and Y-axis as follows: Based on Newton s fist law F x = fo constant mass. The hoizontal components of X-axis ae ULM Based on Newton s second law components of Y-axis ae UALM. x = cos α t (1) x = cos α. () F y = ma y fo constant mass the etical y = sin α t.5gt (3) y Substituting (1) into (3) we hae: = sin α gt (4) a y = g. (5)
ESTIMATION MODELS USING MATHEMATICAL CONCEPTS 35 x x y = sin α.5g (6) cos α cos α g = ( tan α) x x (7) cos α y = ax bx. (8) Equation (8) also can be consideed as quadatic function in mathematics we can sole it using mathematical concept as follows: y =. To get maximum distance ( x max ) we sole quadatic function aboe by taking = ax bx = ( a bx) x 1 = o a sin α x = = b g x sin α = xmax = (9) g x1 + x sin α x s = = =.5xmax. (1) g Substituting (1) into (8) we hae maximum height: y max = a tan = α sin α b g sin α g g sin α g cos α sin α g = sin α sin α g g sin α =. (11) g
36 Now we will attempt to deie equations on Y-axis fom quadatic function (paabolic equation) and Newton s fist law of motion. We ecall quadatic function in mathematics if it passes though maximum o minimum point and one point abitay we hae equations as follows: Fom (13) y = c( x x p ) + y p (1) y = c( x xs ) + y (13) max s s + max y = c( x xx + x ) y s s + y = cx cxx + cx ymax. (14) c = y ymax. ( x x s ) Since the tajectoy passes though the cente coodinate we hae Substituting (15) into (14) we hae Equation (16) coesponds to (8) ymax tan θ c = =. (15) ( x ) xs s tan θ y = x + ( tan θ) x. (16) x s tan θ g tan θ = tan α and =. (17) x s cos α If we substitute (1) into (16) we will get equations which coesponds to (3) (4) and (5) by deiing (16) with espect to time t... Semicicula motion Suppose an object is thown obliquely nea the eath s suface with initial elocity and α is an angle between initial elocity ecto and X-axis. We assume that hoizontal X-axis is eath s suface fo small adius.
ESTIMATION MODELS USING MATHEMATICAL CONCEPTS 37 Figue. Semicicula tajectoy. Fom Figue thee is fundamental discepancy between paabolic motion and semicicula motion. If paabolic motion only has an acceleation of gaity on downwad diection while semicicula motion has an acceleation of gaity on downwad diection as well as centipetal acceleation that leads to the cente (ignoing ai fiction) we will sole this poblem using cicle equation as follows: ( x p) + ( y q) =. (18) It is known that the cente of the cicle is ( p ) so (18) becomes ( x p) + y = (19) x px + p + y =. () Since p equals to.5x max and.5x max is the adius then equation () can be witten as follows: x x + y =. (1) Fo a half of cicle in Figue aboe equation (1) can be witten y = x x. () Let we stat analysing equations of motion on X-axis diection to obtain the equations of motion on Y-axis as a function of the time t. Thee ae two possible assumptions we will use: (1) If the equation of motion on the X-axis diection is ULM then Newton s fist law will be alid F x = fo constant mass. But the nomal foce that leads to the cente of the cicle causes a foce on the X-axis diection N x = N cos θ in which
38 N. To defend Newton s fist law is still alid in this assumption it would x equie inetia on the X-axis diection that opposite to N x diection but the magnitude is the same with N x call it ( M x ) so that N x M x =. But inetia M x does not come fom the centifugal foce like a case in hoizontal cicula motion. If M x comes fom centifugal foce the upwad diection on Y-axis will hae pojection of the centifugal foce N y = N sin θ. It is equal to the nomal foce on the downwad diection thus the equations of motion on Y-axis become ULM. It is impossible on etical cicula motion on eath suface. () If the equation of motion on X-axis diection is UALM then Newton s second law will be alid F x = max fo constant mass. We do not need any additional inetia on the X-axis diection because thee is an acceleation a x such that N x = N cos θ = max. The poblem hee if we assume the equation of motion on X-axis diection is UALM equation () will become ey complicated. The second assumption is also not alid because the alue of N is not constant fo eey point so that a x not constant too. Theefoe we will assume that equation on X-axis is ULM with additional inetia M x. Substituting x = cos α into () we hae: y( t) = cos α t ( cos α ). (3) t The bounday condition is dy( t) dt t elocity component on Y-axis is cos α cos α cos α t = y = (4) y and component of acceleation is: d y( t) 4 cos αy ( cos α cos α t ) = a. y = (5) 3 dt 4y The bounday conditions fo elocity and acceleation ae < t <. cos α In mathematics semicicula tajectoy also can be expessed as follows:
ESTIMATION MODELS USING MATHEMATICAL CONCEPTS 39 s = θ (6) with s is the length of tajectoy o the distance taelled on the peiphey of the obit. π θ = ωt with ω = (7) t dθ dt = ω = π adian time fo a half of ound = (8) π = ω = (9) t a s dθ = = = ω. (3) dt Linea speed in (9) can be expessed as follows: = x + y. (31) Fom equation (31) if is not constant then tangential acceleation will not equal to zeo and centipetal acceleation in equation (3) will change fom point to point. It makes elocity components on X-axis ( x ) o Y-axis ( y ) changes too. On the othe hand if is constant then tangential acceleation will equal to zeo. But theoetically fom the esults of semicicula motion in Figue linea speed will not be constant because the elocity component on Y-axis diection is not constant. Now we will eiew the equations of motion in Figue using Newton s fist and second laws of motion. On Y-axis diection we use Newton s second law as follows: F y = ma y. (3) The nomal foce (N) at eey point always leads to the cente while the weight of the object is always downwad on Y-axis diection. By taking any point A abitay in Figue we expess the net foce on Y-axis using equation (3) as follows: N cos( 9 θ) mg = ma (33) A y N y mg = ma y (34) N y + mg a y =. (35) m
4 Fom equation (35) we can say that it is not a constant alue due to the nomal foce N changes fom point to point. The negatie sign states that the diection of the foces ae downwad. This esult coesponds to equation (5) which states that changes fo eey point. Integating with espect to time t on equation (35) we hae: y a y also N y + mg = a ydt = dt (36) m y We hae a adial foce at point A as follows: Substituting (39) into (37) we hae: N A cos( 9 θ) = a ydt = gt dt. (37) m m N A + mg cos( 9 θ) = (38) m N A = mg cos( 9 θ). (39) cos( 9 θ) t y = C gt( 1 cos ( 9 θ)) (4) cos( 9 θ) t y = C gt( sin ( 9 θ)). (41) By enteing initial condition t = we hae: y cos( 9 θ) = sin α g( sin ( 9 ) + θ t. (4) Equation fo position can be witten as follows: y = y dt. By enteing initial condition t = we hae cos( 9 θ) y = sin αt.5 g( sin ( 9 ) + θ t. (43) We hae assumed that the equation of motion on the X-axis diection is ULM so
ESTIMATION MODELS USING MATHEMATICAL CONCEPTS 41 substituting x = cos α t into (43) cos( 9 ) g( sin ( 9 )) y ( tan ) x.5 θ θ = α + x. (44) cos cos α α 3. Results and Discussion Paabolic On X-axis (ULM): x = cos α t x = cos α On Y-axis (UALM): y = sin α t.5gt y = sin α gt a y = g Tajectoy equation on two-dimensional axis : g y = ( tan α) x x α Semicicula On X-axis (ULM): x = cos α t x = cos α On Y-axis cos( 9 θ) y = sin αt.5 + ( sin ( 9 θ) t With bounday condition: y cos( 9 θ) = sin α + g( sin ( 9 θ) t N y + mg ay = m With bounday condition fo elocity and acceleation Tajectoy equation on two-dimensional axis cos( 9 ) g( sin ( 9 )) y ( tan ) x.5 θ θ = α + x cos cos α α Based on the esults aboe a semicicula motion is combinations of ULM on X-axis diection and a motion with changing acceleation on Y-axis. While the paabolic motion is combinations of ULM on the X-axis diection and UALM on the Y-axis. Semicicula motion can be iewed as a special case of paabolic motion whee the coefficient of highest ode changes fom time to time. While the paabolic motion the coefficient of highest ode is constant.
4 The unique case on the table aboe Tajectoy equation on two-dimensional axis fo semicicula motion is simila with paabolic but it is not diffeentiable if adius is zeo. While in mathematics paabolic must be diffeentiable fo eey point abitay. 4. Conclusion Seeal impotant points can be infeed fom the esults aboe: 1. Semicicula motion is a combination of ULM on the diection of the X-axis and a motion with changing acceleation on the Y-axis while the paabolic motion is a combination of ULM on the diection of the X-axis and UALM on the Y-axis.. Semicicula motion can be iewed as a special case of paabolic motion whee the coefficient of highest ode changes fom time to time while the coefficient of highest ode in paabolic motion is constant. 3. If an object is thown upwad obliquely nea the eath s suface and tajectoy we want is a semicicula it equies: a. on X-axis the equations should be ULM b. on Y-axis the equations should be a motion with changing acceleation c. thee is additional inetia M x against nomal foce N x. 4. The equation of motion on the Y-axis using mathematical concepts and Newton s laws can be iewed as a egession non linea in statistics with the geneal fom as follows: Y = Yˆ + e e ~ N( V ) whee Y = Ideal models epesented by a cicle and a paabolic equations Yˆ = Estimation model using Newton s laws of motion e = annoying factos o some othe foces that we did not conside in this model. Position: Fo a paabolic motion (on eath s suface): y = tan θ cos αt + ( tan θ) cos αt xs ˆ y = sin αt.5gt. Velocity:
ESTIMATION MODELS USING MATHEMATICAL CONCEPTS 43 y tan θ = ( tan θ) cos α cos αt x s ˆ y = sin α gt. Acceleation: a y tan θ = cos α xs aˆ y = g. Fo a semicicula motion (on eath s suface): Position: y = cos αt ( cos αt ) cos( 9 θ) ˆ y = sin αt.5 g( sin ( 9 ) + θ t. Velocity: y = cos α cos αt y cos( 9 θ) ˆ y = sin α g( sin ( 9 ) + θ t. Acceleation: a y = 4 cos αy ( cos α cos αt ) 3 4y aˆ y N y + mg =. m 5. If we thow an object upwad obliquely nea the eath s suface the most possible of tajectoy is paabolic. Refeences [1] D. Halliday and R. Resnick Fisika:jilid 1 Tanslated by Pantu Silaban and Ewin Sucipto Elangga Jakata 1978.
44 [] D. Halliday R. Resnick and J. dan Walke Fundamental of Physics John Wiley & Sons New Yok 8. [3] A. Jeffey and H. Dai Handbook of Mathematical Fomulas and Integals Elseie New Yok 8. [4] Pucell J. Edwin and D. Vabeg Kalkulus dan Geometi Analitik: jilid 1 Tanslated by I. Nyoman Susila Bana Katasasmita and Rawuh Elangga Jakata 3. [5] Walpole and E. Ronald et al. Pobability & Statistics fo Enginees & Scientists 8th ed. Pentice Hall New Jesey 7.