Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables


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1 The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long curve C prmetrized by function f, where f : R R n. Hence we hve x(t f(t, or, more simply, x f(t. For us, n will lwys be 2 or 3, but there re physicl situtions in which it is resonble to hve lrger vlues of n, most of wht we do in this section will pply to those cses eqully well. This is lso good time to introduce the Leibniz nottion for derivtive, thus writing dx Df(t. (2.3. At given time t 0, the vector x(t 0 + h x(t 0 represents the mgnitude direction of the chnge of position of the prticle long C from time t 0 to time t 0 + h, s shown in Figure Dividing by h, we obtin vector, x(t 0 + h x(t 0, (2.3.2 h with the sme direction, but with length pproximting the verge speed of the prticle over the time intervl from t 0 to t 0 + h. Assuming differentibility tking the limit s h pproches 0, we hve the following definition. x( t 0 x ( t + h 0 t 0 x( t 0 + h  x( Figure 2.3. Motion long curve C Copyright c by Dn Sloughter 200
2 2 Motion Along Curve Section 2.3 Definition Suppose x(t is the position of prticle t time t moving long curve C in R n. We cll the velocity of the prticle t time t we cll the speed of the prticle t time t. Moreover, we cll v(t d x(t (2.3.3 s(t v(t (2.3. (t d v(t (2.3.5 the ccelertion of the prticle t time t. Exmple is Consider prticle moving long n ellipse so tht its position t ny time t x (2 cos(t, sin(t. Then its velocity is its speed is its ccelertion is For exmple, t t s we hve v ( 2 sin(t, cos(t, sin 2 (t + cos 2 (t 3 sin 2 (t +, ( 2 cos(t, sin(t. x v s ( 2, 2, ( 2,, 2 5 2, ( 2,. 2 See Figure Notice tht, in this exmples, x for ll vlues of t.
3 Section 2.3 Motion Along Curve 3 v 0.5 x Figure Position, velocity, ccelertion vectors for motion on n ellipse Curvture Suppose x is the position, v is the velocity, s is the speed, is the ccelertion, t time t, of prticle moving long curve C. Let T (t be the unit tngent vector N(t be the principl unit norml vector t x. Now T (t dx dx v v v s, (2.3.6 so v s T (t. (2.3.7 Thus Since we hve dv d ds st (t ds N(t T (t + sdt (t (2.3.8 DT (t DT (t, (2.3.9 T (t + s DT (t N(t. (2.3.0 Note tht (2.3.0 expresses the ccelertion of prticle s the sum of sclr multiples of the unit tngent vector the principl unit norml vector. Tht is, T T (t + N N(t, (2.3. where T ds (2.3.2 N s DT (t. (2.3.3
4 Motion Along Curve Section 2.3 However, since T (t N(t re orthogonl unit vectors, we lso hve T (t ( T T (t + N N(t T (t T (T (t T (t + N (T (t N(t (2.3. T N(t ( T T (t + N N(t N(t T (T (t N(t + N (N(t N(t N. (2.3.5 Hence T is the coordinte of in the direction of T (t N is the coordinte of in the direction of N(t. Thus (2.3.0 writes the ccelertion s sum of its component in the direction of the unit tngent vector its component in the direction of the principl unit norml vector. In prticulr, this shows tht the ccelertion lies in the plne determined by T (t N(t. Moreover, T is the rte of chnge of speed, while N is the product of the speed s DT (t, the mgnitude of the rte of chnge of the unit tngent vector. Since T (t for ll t, DT (t reflects only the rte t which the direction of T (t is chnging; in other words, DT (t is mesurement of how fst the direction of the prticle moving long the curve C is chnging t time t. If we divide this by the speed of the prticle, we obtin strd mesurement of the rte of chnge of direction of C itself. Definition Given curve C with smooth prmetriztion x f(t, we cll κ DT (t s(t (2.3.6 the curvture of C t f(t. Using (2.3.6, we cn rewrite (2.3.0 s ds T (t + s2 κn(t. (2.3.7 Hence the coordinte of ccelertion in the direction of the tngent vector is the rte of chnge of the speed the coordinte of ccelertion in the direction of the principl norml vector is the squre of the speed times the curvture. Thus the greter the speed or the tighter the curve, the lrger the size of the norml component of ccelertion; the greter the rte t which speed is incresing, the greter the tngentil component of ccelertion. This is why drivers re dvised to slow down while pproching curve, then to ccelerte while driving through the curve. Exmple is given by Suppose prticle moves long line in R n so tht its position t ny time t x tw + p,
5 Section 2.3 Motion Along Curve 5 where w 0 p re vectors in R n. Then the prticle hs velocity v dx w speed s w, so the unit tngent vector is T (t v s Hence T (t is constnt vector, so DT (t 0 w w. κ DT (t s 0 for ll t. In other words, line hs zero curvture, s we should expect since the tngent vector never chnges direction. Exmple Consider prticle moving long circle C in R 2 of rdius r > 0 center (, b, with its position t time given by x (r cos(t +, r sin(t + b. Then its velocity, speed, ccelertion re s v ( r sin(t, r cos(t, r 2 sin ( t + r 2 cos 2 (t r ( r cos(t, r sin(t, respectively. Hence the unit tngent vector is T (t v s ( sin(t, cos(t. Thus DT (t ( cos(t, sin(t DT (t cos 2 (t + sin 2 (t. Hence the curvture of C is, for ll t, κ DT (t s r.
6 6 Motion Along Curve Section 2.3 Thus circle hs constnt curvture, nmely, the reciprocl of the rdius of the circle. In prticulr, the lrger the rdius of circle, the smller the curvture. Also, note tht so, from (2.3.0, we hve ds d r 0, rn(t, which we cn verify directly. Tht is, the ccelertion hs norml component, but no tngentil component. Exmple time t is Now consider prticle moving long n ellipse E so tht its position t ny x (2 cos(t, sin(t. Then, s we sw bove, the velocity speed of the prticle re v ( 2 sin(t, cos(t s 3 sin( 2 +, respectively. For purposes of differentition, it will be helpful to rewrite s s cos(2t s ( cos(2t Then the unit tngent vector is 2 T (t ( 2 sin(t, cos(t. 5 3 cos(2t Thus 2 DT (t 5 3 cos(2t So, for exmple, t t, we hve 3 2 sin(2t ( 2 cos(t, sin(t ( 2 sin(t, cos(t. (5 3 cos(2t 3 2 x v s ( 2, 2, ( 2,, 2 5 2,
7 Section 2.3 Motion Along Curve 7 ( T ( DT 5 ( 2,, 5 (, 8, 5 ( DT ( 2, Hence the curvture of E t is 2 κ , where the finl numericl vlue hs been rounded to four deciml plces. Although the generl expression for κ is complicted, it is esily computed plotted using computer lgebr system, s shown in Figure Compring this with the plot of this ellipse in Figure 2.3.2, we cn see why the curvture is gretest round (2, 0 ( 2, 0, corresponding to t 0, t, t 2, smllest t (0, (0,, corresponding to t 2 t 3 2. Finlly, s we sw bove, the ccelertion of the prticle is so Now if we write ( 2 cos(t, sin(t, then we my either compute, using (2.3.7, T ds or, using (2.3. (2.3.5, T ( 2,. 2 T T (t + N N(t, 2 (5 3 cos(2t 2 (3 sin(2t N s 2 k ( T , 5 0 ( 2, ( 2, N ( N ( 2, 2 5 (, 8 0.
8 8 Motion Along Curve Section Figure Curvture of n ellipse Hence, in either cse, 3 ( T + ( N. 0 0 Arc length Suppose prticle moves long curve C in R n so tht its position t time t is given by x f(t let D be the distnce trveled by the prticle from time t to t b. We will suppose tht s(t v(t is continuous on [, b]. To pproximte D, we divide [, b] into n subintervls, ech of length t b n, lbel the endpoints of the subintervls t 0, t,..., t n b. If t is smll, then the distnce the prticle trvels during the jth subintervl, j, 2,..., n, should be, pproximtely, s t, n pproximtion which improves s t decreses. Hence, for sufficiently smll t (equivlently, sufficiently lrge n, n s(t j t (2.3.8 j will provide n pproximtion s close to D s desired. Tht is, we should define D lim n j n s(t j t. (2.3.9 But (2.3.8 is Riemnn sum (in prticulr, lefth rule sum which pproximtes the definite integrl b s(t. (2.3.20
9 Section 2.3 Motion Along Curve 9 Hence the limit in (2.3.9 is the vlue of the definite integrl (2.3.20, so we hve the following definition. Definition Suppose prticle moves long curve C in R n so tht its position t time t is given by x f(t. Suppose the velocity v(t is continuous on the intervl [, b]. Then we define the distnce trveled by the prticle from time t to time t b to be b v(t. (2.3.2 Note tht the distnce trveled is the length of the curve C if the prticle trverses C exctly once. In tht cse, we cll (2.3.2 the length of C. In generl, for ny t such tht the intervl [, t] is in the domin of f, we my clculte σ(t which we cll the rc length function for C. Exmple t Consider the helix H prmetrized by f(t (cos(t, sin(t, t. v(u du, ( If we let L denote the length of one complete loop of the helix, then prticle trveling long H ccording to x f(t will trverse this distnce s t goes from 0 to 2. Since we hve Hence Exmple by v(t v(t ( sin(t, cos(t,, L sin 2 (t + cos 2 (t Suppose prticle moves long curve C so tht its position t time t is given x (( + 2 cos(t cos(t, ( + 2 cos(t sin(t. Then C is the curve in Figure 2.3., which is clled limçon. The prticle will trverse this curve once s t goes from 0 to 2. Now so v ( ( + 2 cos(t sin(t 2 sin(t cos(t, ( + 2 cos(t cos(t 2 sin 2 (t, v 2 v v ( + 2 cos(t 2 sin 2 (t + ( + 2 cos(t sin 2 (t cos(t + sin 2 (t cos 2 (t + ( + 2 cos(t 2 cos 2 (t ( + 2 cos(t sin 2 (t cos(t + sin (t ( + 2 cos(t 2 (sin 2 (t + cos 2 (t + sin 2 (t cos 2 (t + sin (t ( + 2 cos(t 2 + sin 2 (t cos 2 (t + sin (t,
10 0 Motion Along Curve Section Figure 2.3. A limçon Hence the length of C is 2 0 ( + 2 cos(t 2 + sin 2 (t cos 2 (t + sin (t 3.369, where the integrtion ws performed with computer the finl result rounded to four deciml plces. Note tht integrting from 0 to would find the distnce the prticle trvels in going round C twice, nmely, 0 ( + 2 cos(t 2 + sin 2 (t cos 2 (t + sin (t Problems. For ech of the following, suppose prticle is moving long curve so tht its position t time t is given by x f(t. Find the velocity ccelertion of the prticle. ( f(t (t 2 + 3, sin(t (b f(t (t 2 e 2t, t 3 e 2t, 3t (c f(t (cos(3t 2, sin(3t 2 (d f(t (t cos(t 2, t sin(t 2, 3t cos(t 2 2. Find the curvture of the following curves t the given point. ( f(t (t, t 2, t (b f(t (3 cos(t, sin(t, t
11 Section 2.3 Motion Along Curve (c f(t (cos(t, sin(t, t, t 3 (d f(t (cos(t, sin(t, e t, t 0 3. Plot the curvture for ech of the following curves over the given intervl I. ( f(t (t, t 2, I [ 2, 2] (b f(t (cos(t, 3 sin(t, I [0, 2] (c g(t (( + 2 cos(t cos(t, ( + 2 cos(t sin(t, I [0, 2] (d h(t (2 cos(t, sin(t, 2t, I [0, 2] (e f(t ( cos(t + sin(t, sin(t + sin(t, I [0, 2]. For ech of the following, suppose prticle is moving long curve so tht its position t time t is given by x f(t. Find the coordintes of ccelertion in the direction of the unit tngent vector in the direction of the principl unit norml vector t the specified point. Write the ccelertion s sum of sclr multiples of the unit tngent vector the principl unit norml vector. ( f(t (sin(t, cos(t, t 3 (b f(t (cos(t, 3 sin(t, t (c f(t (t, t 2, t (d f(t (sin(t, cos(t, t, t 3 5. Suppose prticle moves long curve C in R 3 so tht its position t time t is given by x f(t. Let v, s, denote the velocity, speed, ccelertion of the prticle, respectively, let κ be the curvture of C. ( Using the fcts v st (t ds T (t + s2 κn(t, show tht v s 3 κ(t (t N(t. (b Use the result of prt ( to show tht κ v v Let H be the helix in R 3 prmetrized by f(t (cos(t, sin(t, t. Use the result from Problem 5 to compute the curvture κ of H for ny time t. 7. Let C be the ellipticl helix in R 3 prmetrized by f(t ( cos(t, 2 sin(t, t. Use the result from Problem 5 to compute the curvture κ of C t t. 8. Let C be the curve in R 2 which is the grph of the function ϕ : R R. Use the result from Problem 5 to show tht the curvture of C t the point (t, ϕ(t is κ ϕ (t. ( + (ϕ (t 2 3 2
12 2 Motion Along Curve Section Let P be the grph of f(t t 2. Use the result from Problem 8 to find the curvture of P t (, (2,. 0. Let C be the grph of f(t t 3. Use the result from Problem 8 to find the curvture of C t (, (2, 8.. Let C be the grph of g(t sin(t. Use the result from Problem 8 to find the curvture of C t ( ( 2,, For ech of the following, suppose prticle is moving long curve so tht its position t time t is given by x f(t. Find the distnce trveled by the prticle over the given time intervl. ( f(t (sin(t, 3 cos(t, I [0, 2] (b f(t (cos(t, sin(t, 2t, I [0, ] (c f(t (t, t 2, I [0, 2] (d f(t (t cos(t, t sin(t, I [0, 2] (e f(t (cos(2t, sin(2t, 3t 2, t, I [0, ] (f f(t (e t cos(t, e t sin(t, I [ 2, 2] (g f(t ( cos(t + sin(t, sin(t + sin(t, I [0, 2] 3. Verify tht the circumference of circle of rdius r is 2r.. The curve prmetrized by f(t (sin(2t cos(t, sin(2t sin(t hs four petls. Find the length of one of these petls. 5. The curve C prmetrized by h(t (cos 3 (t, sin 3 (t is clled hypocycloid (see Figure in Section 2.3. Find the length of C. 6. Suppose ϕ : R R is continuously differentible let C be the prt of the grph of ϕ over the intervl [, b]. Show tht the length of C is b + (ϕ (t Use the result from Problem 6 to find the length of one rch of the grph of f(t sin(t. 8. Let h : R R n prmetrize curve C. We sy C is prmetrized by rc length if Dh(t for ll t. ( Let σ be the rc length function for C using the prmetriztion f let σ be its inverse function. Show tht the function g : R R n defined by g(u f(σ (u prmetrizes C by rc length.
13 Section 2.3 Motion Along Curve 3 (b Let C be the circulr helix in R 3 with prmetriztion f(t (cos(t, sin(t, t. Find function g : R R n which prmetrizes C by rc length. 9. Suppose f : R R n is continuous on the closed intervl [, b] hs coordinte functions f, f 2,..., f n. We define the definite integrl of f over the intervl [, b] to be ( b b b b f(t f (t, f 2 (t,..., f n (t. Show tht if prticle moves so its velocity t time t is v(t, then, ssuming v is continuous function on n intervl [, b], the position of the prticle for ny time t in [, b] is given by x(t t v(sds + x(. 20. Suppose prticle moves long curve in R 3 so tht its velocity t ny time t is v(t (cos(2t, sin(2t, 3t. If the prticle is t (0,, 0 when t 0, use Problem 9 to determine its position for ny other time t. 2. Suppose prticle moves long curve in R 3 so tht its ccelertion t ny time t is (t (cos(t, sin(t, 0. If the prticle is t (, 2, 0 with velocity (0,, t time t 0, use Problem 9 to determine its position for ny other time t. 22. Suppose projectile is fired from the ground t n ngle α with n initil speed v 0, s shown in Figure Let x(t, v(t, (t be the position, velocity, ccelertion, respectively, of the projectile t time t. v(0 α Figure The pth of projectile
14 Motion Along Curve Section 2.3 ( Explin why x(0 (0, 0, v(0 (v 0 cos(α, v 0 sin(α, (t (0, g for ll t, where g 9.8 meters per second per second is the ccelertion due to grvity. (b Use Problem 9 to find v(t. (c Use Problem 9 to find x(t. (d Show tht the curve prmetrized by x(t is prbol. Tht is, let x(t (x, y show tht y x 2 + bx + c for some constnts, b, c. (e Show tht the rnge of the projectile, tht is, the horizontl distnce trveled, is R v 0 sin(2α g conclude tht the rnge is mximized when α. (f When does the projectile hit the ground? (g Wht is the mximum height reched by the projectile? When does it rech this height? 23. Suppose, 2,..., m re unit vectors in R n, m n, which re mutully orthogonl (tht is, i j when i j. If x is vector in R n with show tht x i x i, i, 2,..., m. x x + x x m m,
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