GUARDIAN 19. At an air show, a jet has a trajectory following the curve y = x2. ANDITSNUT daniAV SECTION 14.3 / Arc Length and Speed 765 If when the jet is at the point (1, 1), it has a speed of 500 km/h, determine its velocity vector at this point. 27. Let r(1) = (31 + 1,4t - 5,21). 20 . What is the velocity vector of a particle traveling to the right along the hyperbola y = x" with constant speed 5 cm/s when the particle's location (a) Evaluate the arc length integral s = (1) = Ir()udu. na is ( 2, 2 ) ? b) Find the inverse 8-1(s) of 8 (1). 21 . A bee with velocity vector r'(1 ) starts out at the origin at 1 = 0 (e) Verify that ri(s) = r(g-1(s)) is an arc length parametrization. and flies around for 7 seconds. Where is the bee located at time ?' if 28. Find an arc length parametrization of the line y = 4x +9. "(u) du = 0? What does the quantity Ir'(u) Il du represent? 29. Let r(1) = w + tv be a parametrization of a line. 22. The DNA molecule comes in the form of a double helix, meaning two (a) Show that the arc length function s = (1) = Ir(1)ll du is given helices that wrap around one another. Suppose a single one of the helices by s = filvil. This shows that r(f) is an arc length parametrization if and has a radius of 10A (1 angstrom A = 10-8 cm) and one full turn of the helix has a height of 34A. only if v is a unit vector. V = (3, 4, 5). (b) Find an arc length parametrization of the line with w = (1,2,3) and helix . (a ) Show that r(f) = (10 cost, 10 sint, 27 ) is a parametrization of the 30. Find an arc length parametrization of the circle in the plane z = 9 with b) Find the arc length of one full turn of the helix. radius 4 and center (1, 4,9). 23. Let 31. Find a path that traces the circle in the plane y = 10 with radius 4 and center (2, 10, -3) with constant speed 8. I(1) = (RCOS (27 NI " h" ) , Resin ( 2 7 NI ) 32. Find an arc length parametrization of the curve parametriza osish r(t) = (t, 313/2, 2-13/2 ), with the parameter s measuring from (0, 0, 0). on ai (a) Show that r(t) parametrization radius R and height h making 33. Find an arc length parametrization of the curve parametriza N complete turns. r(t) = (cost, sint, 313/2), with the parameter s measuring from (1, 0, 0). (b) Guess which of the two springs in Figure 6 uses more wire. 34. Find an arc length parametrization of the curve parametriza c) Compute the lengths of the two springs and compare. r(t) = (e' sint, et cost, et). 35. Find an arc length parametrization of the curve parametriza I(t) = (12, 13). 36. Find an arc length parametrization of the cycloid with parametrization r(t) = (t - sint, 1 - cost). 37. Find an arc length parametrization of the line y = mx for an arbitrary 3.em slope m. + cm 38. Express the arc length L of y = x3 for 0
