Tarzan s Dilemma for Elliptic and Cycloidal Motion

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1 Tarzan Dilemma or Elliptic and Cycloidal Motion Yuji Kajiyama National Intitute o Technology, Yuge College, Shimo-Yuge 000, Yuge, Kamijima, Ehime, , Japan kajiyama@gen.yuge.ac.jp btract-in thi paper, we tudy Tarzan dilemma o elliptic and cycloidal motion. We give the relation between the lying ditance and the launching angle o thee two dierent motion, and perorm numerical calculation to ind the launching angle which maximize the lying ditance. Thi tudy will be helpul or undertanding elementary mechanic or tudent. Keyword- Claical Mechanic; Phyic Education I. INTRODUCTION When one throw a ball rom the ground into the air with deinite peed and launching angle, what i the launching angle to maximize the lying ditance? The anwer o thi well-known problem can be derived rom imple calculation o Newtonian mechanic. Next, let u conider a ituation that an object (aying Tarzan ) i attached at one end o rope and the other end i ixed at a point above the ground. I Tarzan tart winging due to the gravity and releae hi hand rom the rope at an intermediate point, he i thrown into the air with correponding launching angle. The problem i that what angle give the maximal lying ditance. I Tarzan want to get large 45 (cloed to 45 ), the launching peed i low becaue he loe kinetic energy. I he want to get large launching peed, he cannot get enough large launching angle. Thereore thi problem i called Tarzan dilemma [ 5]. In previou work [ 5], the author have conidered the cae o ordinary circular motion. However, thi i not the only way to jump when Tarzan ha a rope o deinite length. I both end o rope are ixed at each point above the ground, and Tarzan hold midpoint o the rope, hi trajectory o the wing will be an ellipe, horizontally or vertically long ellipe. I one end o the rope i ixed at a point between two inverted cycloidal object, Tarzan trajectory will be a cycloidal curve. In any cae, one can ind the launching angle which maximize the lying ditance L and compare that what way o wing will be the bet or Tarzan. In thi paper, we conider Tarzan dilemma in elliptic and cycloidal motion and perorm numerical calculation to ind the bet launching angle, comparing to thoe o ordinary circular motion. In educational point o view, olving thi problem develop tudent qualitative and conceptual undertanding or mechanical motion and energy conervation law without tough calculation. For example, what i the bet way o wing to reach the arthet ditance and what i the magnitude relation o between three way? In the paper, we dicu the cae o elliptic motion and cycloidal motion and derive a ormula or the lying ditance L in analogy with the cae o circular motion. Beide, we dicu the motion o Tarzan in detail by uing the reult o previou ection. Comparing our way o winging, we ind that the lying ditance i the larget in the cae o circular motion when the length o rope i deinite. II. LIPTIC TRZN We dicu Tarzan jump in the cae o elliptic motion. hown in the let panel o Figure, let u conider the ituation that Tarzan ha a rope o the length a, and ixe it at two point F and F with the ditance a b, with a > b. He tand at the tarting point holding the midpoint o the rope and tart to wing. I riction between the rope and hi hand i negligible, he move along an elliptic curve with emi-major (emi-minor) axi a (b) due to the gravity and tenion rom the rope. ter paing the lowet point B o the height h, he lie into the air by releaing the rope at point C, and inally land on point D. Now we reinterpret thi ituation a wing with an imaginary rope ixed at one point, a hown in the right panel o Figure. Thi imaginary rope change it length uch that Tarzan can move along elliptic curve. While it i diicult to dicu tenion o uch rope, one doe not have to worry about it becaue only geometrical conideration i needed or calculation. lthough we dicu the cae o a > b with eccentricity e ( a b ) / a, the ollowing dicuion are valid or the cae o a < b a well by uing e ( b a ) / b. In the elliptic cae, the length o the elatic rope varie a a unction o poition unlike the ordinary circular cae. Tarzan tart winging rom the tarting point with the poition x r in, y h b r co (.) DOI: /JBP

2 For t t t r, the poition o Tarzan in the air i given by e r a. (.) e in x( t) r in v t co, (.3) y( t) h b r co gt vt in, (.4) i deined imilar to Eq. (.) by replacing to the angle o the launching point C, acceleration. The angle t t t and the launching angle tan i related a, Tarzan land on the ground o point D with the poition. g i gravitational b tan. (.5) a L r in v t co, (.6) 0 h b r co vt in gt From Eq. (.6), (.7) and the energy conervation law between point and point C we obtain the lying ditance L rom the origin a a unction o. (.7) mg[ h b r co] mv mg[ h b r co], (.8) a in 4a co h b r co L r in a in, (.9) r co r co, (.0) a correpond to the dierence in height between point and point C. One can check that in the cae o a = b(= r), Eq. (.9) reduce to that o the circular cae []. The problem i to ind the value o. The above dicuion are valid or the cae o a < b by replacing r, to r, which maximize L or given a, b, h and deined a r a, r a. (.) e co e co We perorm numerical calculation in ection 5 or both cae a > b (the cae -) and a < b (the cae -). Figure Tarzan jump o elliptic motion. Let: Tarzan wing with the rope o length a which i ixed at two point F and F. Right: Tarzan jump with an elatic rope ixed at one point. We perorm calculation and analyze the motion in the notation o the right panel or convenience. DOI: /JBP

3 Figure Tarzan jump o cycloidal motion. Let: Tarzan wing with the rope o length 4R, which i ixed at a point between two cycloidal object. Right: Equivalent igure o Tarzan jump with an imaginary elatic rope whoe length can be changed uch that Tarzan move along the cycloidal curve. III. CLOIDL TRZN the next example o Tarzan wing, we dicu the cae o cycloidal wing o Tarzan in thi ection. hown in the let panel o Figure, i a rope o the length 4R i ixed at a point o the height h + 4R between two inverted cycloidal object generated rom a circle with radiu R, Tarzan move along cycloidal curve while winging. The parametric orm o the reulting cycloid o Tarzan i given by x R( in ), y h R( co ), (3.) with 0. gain we reinterpret thi cycloidal motion a wing with an imaginary elatic rope ixed at a point o the height H = h+r a depicted in the right panel o Figure. The length o the rope can be changed rom R to end o the rope (Tarzan) move along the cycloidal curve. We deine angle or 0 and or meauring rom the y-axi, a hown in the Figure. The angle and correponding to point and point C are related to the angle and a given below. Tarzan tart winging rom point with the poition ter launching rom point C with the angle and t t t x in tan co R uch that it r in, y h R r co, (3.), at t t in co, r R. (3.3), Tarzan in the air, or t t t, i at the poition x( t) r in v t co, (3.4) y( t) h R r co gt vt in, (3.5) in tan, in co r R, (3.6) co, Tarzan land on the ground at point D with the poition tan cot (3.7) L r in v t co, (3.8) DOI: /JBP

4 . (3.9) 0 h R r co vt in gt The energy conervation law between point and point C i given by mg[ h R r co ] mv mg[ h R r co ], (3.0) a uual. From Eq. (3.8), (3.9) and (3.0), we obtain the lying ditance L rom the origin a R in 4R co h R r co L r in R in, (3.) r co r co, (3.) R ha been deined a the cae o the previou ection. In the next ection, we perorm numerical calculation to ind the value o which maximize L. We call the motion o thi type the cae. Figure 3 Tarzan trajectory or CI (black olid), - (blue-dahed), - (green dot-dahed), and (red thick). Three horizontal dahed line correpond to H(), M() and L() rom top to bottom. IV. DISCUSSIONS In thi ection, we dicu relation between the lying ditance L and the launching angle in the cae given in the previou ection, uch a circular (CI) [], elliptic (- and -) and cycloidal () Tarzan. We perorm numerical calculation and compare dierence between wing under the ollowing condition. () The length o rope i 5.0 m. We et r = 5.0 m (CI), (a, b) = (.5,.0) m (-), (a, b) = (.0,.5) m (-), and R =.5 m (). () When the tarting angle = 90 [deg.], the tarting point i H = 0 (or 5.0) m high. For comparion, we et the height o point to be (0, 9., 8.3) m reerring to cae (H, M, L), and (5.0, 4., 3.3) m reerring to cae (H, M, L). Relative height and trajectory are hown in Figure 3. For the cae L and L in -, there i no olution becaue the tarting point i below the lowet point B. DOI: /JBP

5 Figure 4 how the relation between the lying ditance L and the launching angle or H = 0 m (upper three panel) and H = 5.0 m (lower three panel). In each panel, black olid, blue dahed, green dot-dahed and red thick curve correpond to CI, -, - and, repectively. Since the height h o the lowet point B i the ame or - and, green dot-dahed curve and red thick curve croe at = 0 in any cae. One can ee that the CI cae give the longet ditance under any condition becaue Tarzan can get the larget kinetic energy while winging. For H = 5.0 m cae (H, M and L), the lying ditance L o CI cae i maller than that o other cae or relatively mall unlike H = 0 m cae (H, M and L), becaue Tarzan touche the ground at the lowet point B ( = 0) and thereore he cannot get large L unle become large. Obviouly each curve ha the maximal value. The maximal value o the lying ditance L and the correponding launching angle are given in Table or all cae o Figure 4. In each element o Table, the value ( [deg.], [deg.], L[m]) are given, L i the maximal value or given height o point and i the correponding launching angle. hown in Table, the launching angle to get the longet L become maller when the tarting point become lower. The reaon i that a the point become lower, larger horizontal peed time t t v co i needed to get longer lying ditance L and the lying become horter. Comparing the our cae CI, -, - and, a the lowet point B i lower, the lying ditance L become larger becaue Tarzan can get larger kinetic energy. However or - and with the ame height o point B, the maximal value o L or i alway larger than that o - becaue the x-coordinate or i larger than that o -, depending on curvature. or the launching angle, the magnitude relation between our way i x t ) ( CI) ( ) ( ) ( ) (4.) in any cae. Thi relation depend on what kinetic energy Tarzan can get. Thi can be een in Figure 5. Figure 5 i the trajectory o Tarzan or the cae o = 90 [deg.]. The trajectorie o - and are imilar becaue the lowet point B i the ame in height in both cae. The maximal value o the lying ditance L trongly depend on the height h o the lowet point B. However, it depend on and type o the wing a well. The general dicuion o dependence o L on h i beyond the cope o thi manucript. In the educational point o view, thi problem require conceptual undertanding o phyic or tudent a well a ome calculation. Student can derive the relation between L and without tough calculation, only by law o projectile moving due to the gravity and energy conervation law. lthough the dependence o L eem complicated, tudent can ind that there exit the maximal value o L and dicu the reaon o the relation Eq. (4.) given above. Conidering and dicuing uch phyical meaning will be helpul or tudent to undertand phyic. Table The value ( [deg.], [deg.], L [m]) or deinite height o point. L i the maximal value or given height o point, and i the correponding launching angle. Height o [m] CI - - 0(H) (90, 5.4,.) (90,.9, 9.58) (90, 9.4, 4.) (90, 8.5, 5.0) 9.(M) (80, 3.8, 8.9) (55.0, 4.4, 3.4) (47., 6.,.) (76., 5.4,.9) 8.3(L) (70,.8, 6.5) No (3.,.7, 7.64) (57.,., 8.0) 5.0(H) (90,4.0,.0) (90, 0.8, 5.53) (90, 34.8, 7.) (90, 3., 8.5) 4.(M) (80, 40., 0.) (55.0, 7.8,.94) (47., 3.6, 5.3) (76., 7.3, 6.49) 3.3(L) (70, 39., 8.50) No (3., 5.5, 3.3) (57., 0.8, 4.4) ( DOI: /JBP

6 Figure 4 Flying ditance L a a unction o the launching angle [deg.] or H = 0 m (upper three panel) and H = 5.0 m (lower three panel). In each panel, black olid, blue dahed, green dot-dahed and red thick curve correpond to CI, -, - and, repectively. Figure 5 Tarzan trajectory in the cae o = 90 [deg.] or CI (black), -(blue dahed), - (green dot-dahed), (red thick) motion, or H = 0 m (upper our curve) and H = 5.0 m (lower our curve). The length o the rope i 5.0 m in all cae. DOI: /JBP

7 V. CONCLUSIONS We have tudied Tarzan dilemma in three dierent motion, including circular, elliptic and cycloidal wing. Thee three motion can be realized by ixing rope in dierent way. We have derived ormulae o the lying ditance L a a unction o the launching angle by imple calculation and perormed numerical calculation to ind the maximal value o L and correponding or each type o wing. There exit the value o which maximize L, and it value o increae a the height o the lowet point B decreae, depending on the type o wing. the lowet point B become lower, Tarzan can get larger kinetic energy and thereore become larger (cloe to ). In our calculation, we have et that the cae and - have the ame lowet point in height. In uch a cae, trajectory o Tarzan i imilar with each other, and the dierence come rom curvature o wing. Calculation and phyical interpretation to olve thi Tarzan dilemma will be helpul or tudent to develop conceptual undertanding o phyic a well a quantitative calculation. 45 REFERENCES [] K. P. Trout and C.. Gaton, ctive-learning phyic experiment uing the Tarzan Swing, Phyic Teacher, vol. 39(3), pp , 00. [] H. Shima, How ar can Tarzan jump? European Journal o Phyic, vol. 33, pp , 0. [3] M. Rave and M. Sayer, Tarzan Dilemma: Challenging Problem or Introductory Phyic Student, Phyic Teacher, vol. 5(8), pp , 03. [4] C. E. Mungan, nalytically olving Tarzan Dilemma, Phyic Teacher, vol. 5, pp. 6, 04. [5] W. Klobu, Motion on a vertical loop with riction, merican Journal o Phyic, vol. 79(9), pp , 0. DOI: /JBP

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