Mathematics of Motion II Projectiles
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1 Chmp+ Fll 2001 Dn Stump 1 Mthemtics of Motion II Projectiles Tble of vribles t time v velocity, v 0 initil velocity ccelertion D distnce x position coordinte, x 0 initil position x horizontl coordinte y verticl coordinte Summry of equtions For constnt velocity, v 0 x(t) = x 0 + v 0x t There is constnt downwrd force, due to grvity. This produces constnt negtive ccelertion for the y coordinte y = g where g = 9.8 m/s 2 = 32 ft/s 2. Therefore v y (t) = v 0y gt y(t) = y 0 + v 0y t 1 2 gt2 Picture of the trjectory showing the position of the projectile t equl time intervls. v = v 0 x = x 0 + v 0 t D = v 0 t For constnt ccelertion, v = v 0 + t x = x 0 + v 0 t t2 D = v 0 t t2 y ft x ft Projectile motion Consider motion in Erth s grvity. Exmples re bsebll nd other sports, militry wepons like ctpults or mortrs, nd stellites nd rockets. In the simplest cses there re two coordintes, The trjectory is prbol with negtive curvture. We cn plot trjectory curve by mking prmetric plot; tht is, plot y(t) versus x(t) using t s n independent prmeter. Your grphing clcultor should hve prmetric plot mode. x = horizontl coordinte y = verticl coordinte We ll neglect ir resistnce nd erodynmic forces of drg nd lift. There is no horizontl force so the x component of velocity is constnt v x (t) = v 0x
2 Chmp+ Fll 2001 Dn Stump 2 Exercises (b) x(t) = t, y(t) = 1/t. Exercise 1 A bll is thrown horizontlly t 50 mi/hr from height of 5 ft. Where will it hit the ground? Exercise 2 (c) x(t) = cos(2πt), y(t) = sin(2πt). (d) x(t) = 2 cos(2πt), y(t) = 0.5 sin(2πt). (e) x(t) = cos(2πt/3), y(t) = sin(2πt/7). Exercise 4 Bsebll home run A slugger hits bll. The speed of the bll s it leves the bt is v 0 = 100 mi/hr = 147 ft/sec. Suppose the initil direction is 45 degrees bove the horizontl, nd the initil height is 3 ft. () Plot y s function of x, using Mthemtic. The cstle is 300 m distnt from the ctpult. If the initil direction of the velocity vector of the stone is 45 degrees bove the horizontl, wht initil speed v 0 is required to hit the cstle? (Hint: The initil velocity vector is v 0 = îv 0 cos 45 + ĵv 0 sin 45; tht is, v 0x = v 0 / 2 nd v 0y = v 0 / 2.) Exercise 3 Prmetric plots in Mthemtic A prmetric plot is kind of grph curve of y versus x where x nd y re known s functions of n independent vrible t clled the prmeter. To plot the curve specified by x = f(t) nd y = g(t), the Mthemtic commnd is PrmetricPlot[{f[t],g[t]},{t,t1,t2}, PlotRnge->{{x1,x2},{y1,y2}}, AspectRtio->r] Here {t1,t2} is the domin of t, nd {x1,x2} nd {y1,y2} re the rnges of x nd y. To mke the x nd y xes hve the sme scles, r should hve the numericl vlue of (y2-y1)/(x2-x1). Use Mthemtic to mke the prmetric plots below. ech cse stte in words wht the curve is. () x(t) = t, y(t) = t t 2. In x[t_]:= (put the eqution for x here) y[t_]:= (put the eqution for y here) PrmetricPlot[{x[t],y[t]},{t,0,5}, PlotRnge->{{0,350},{0,100}}, AspectRtio->100/350, AxesLbel->{"x (ft)","y (ft)"}] (b) When precisely does the bll hit the ground? (Hint: Use the Mthemtic commnd FindRoot. To solve the eqution F (t) = C, for t, the Mthemtic commnd is FindRoot[F[t]==C,{t,t1}] where t 1 is n initil estimte of the solution.) (c) Where precisely does the bll hit the ground? (Hint: Just sk Mthemtic for x[the nswer to (b)].) (d) We hve neglected ir resistnce. Is tht good pproximtion? Justify your nswer. Exercise 5 Conservtion of energy for projectile () Consider projectile, moving under grvity but with negligible ir resistnce, such s shot put. Assume these initil vlues x 0 = 0 nd v 0x = 10 m/s y 0 = 1.6 m nd v 0y = 8 m/s Use Mthemtic or grphing clcultor to mke plots of x versus t nd y versus t. Hnd in sketches of the plots, nd remember to show the scles on the xes.
3 Chmp+ Fll 2001 Dn Stump 3 (b) Now plot the totl energy (kinetic plus potentil) versus t, E(t) = 1 2 m ( v 2 x (t) + v2 y (t)) + mgy(t) where m = 7 kg. (c) Prove mthemticlly tht E is constnt of the motion. Exercise 6 The jumping squirrel The squirrel wnts to jump from A to B. The horizontl distnce from A to B is x = 5 ft, nd the verticl distnce is y = 4 ft. If the squirrel jumps with n initil speed of 20 ft/s, t wht ngle to the horizontl should it jump? Exercise 7 The trctrix The trctrix is the curve followed by n object pulled by string nd sliding on frictionl surfce. The instntneous velocity of the object is lwys in the direction of the string, nd the string length remins constnt. The initil position of the object is (x 0, y 0 ) = (0, ); tht is, the object is initilly on the y xis t distnce from the origin. The length of the string is. The other end of the string is initilly t the origin; therefter it moves with constnt velocity v long the x xis. Prmetric equtions for the trctrix, i.e., the trjectory of the pulled object in the xy plne, re ( ) vt x = vt tnh, y = ( ). vt cosh () Plot the trctrix the curve in the xy plne for unit vlues of nd v. (b) When the end of the string is t (2, 0) (with = 1) where is the drgged object?
4 Chmp+ Fll 2001 Dn Stump 4 Exercise 8 The cycloid Consider circle of rdius R tht rolls without slipping on line. For exmple, the circle could be bicycle tire, nd the stright line rod. We ll determine the motion of point P fixed on the circle. For exmple, if there is white dot pinted on the side of the bicycle tire, how does tht white dot move s the tire rolls long the rod? Figure 1 shows the circle t three different times. At t = 0 the point P is t the origin. The circle rolls on the x xis. At t = 1 the point P is t the top of the circle, nd t time t = 2 it is bck t the bottom gin. 1 2 () Use trigonometry to determine the x coordinte of the point P in Figure 2. (The result depends on sin θ.) (b) Use trigonometry to determine the y coordinte of the point P in Figure 2. (The result depends on cos θ.) (c) Mke plot of the curve trced out by the point P in the xy spce. (Set R = 1.) (Hint: From () nd (b) you hve equtions for x nd y s functions of the prmeter θ, x = f(θ) nd y = g(θ), Now consider n rbitrry time t, when the line connecting P to the center of the circle is t ngle θ to the verticl, s shown in Figure 2. At this time t the center of the circle is t x = Rθ: becuse Rθ is the rclength, nd the rc hs the sme length s the distnce trveled long the rod if the circle rolls without slipping. (Note: the ngle θ is mesured in rdins.) so use the prmetric plot mode of your grphing clcultor to plot y versus x.) The cycloid. The curve trced out by P s the circle rolls long the line is clled cycloid. If you re stuck t this point, try looking up cycloid curve on the Internet. (d) From the result of (c) prove tht when P is in contct with the line tht the circle rolls on (i.e., the x xis in Figure 1 or 2) the instntneous velocity of P is 0. In other words, when you ride bicycle, the point on either tire in contct with the rod is instntneously t rest. So how cn you be moving!?
5 Chmp+ Fll 2001 Dn Stump 5 Answers Nov 1, 2001 Exercise 1 Where will the bll hit the ground? (d) Is it good pproximtion? Explin. Exercise 2 Wht initil speed is required to hit the cstle? Exercise 5 Conservtion of energy () Sketch plots of x(t) nd y(t); include scles on the xes. Exercise 3 Prmetric plots () Sketch the grph. (b) Sketch the grph. (c) Sketch the grph. (d) Sketch the grph. (e) Sketch the grph. (b) Sketch plot of E versus t. (c) Proof: Exercise 6 The jumping squirrel At wht ngle should the squirrel jump to rech the other brnch? Exercise 7 The trctrix () Sketch plot of the trctrix. Exercise 4 Bsebll home run () Sketch the trjectory; include scles on the xes. (b) When the end of the string is t (0, 2) where is the drgged object? (b) Time when the bll hits the ground = (c) Distnce where the bll hits the ground =
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