{"id":8286,"date":"2022-07-10T10:39:11","date_gmt":"2022-07-10T15:39:11","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/analisisnumerico\/?p=8286"},"modified":"2026-04-05T20:02:23","modified_gmt":"2026-04-06T01:02:23","slug":"s1eva2022paoi_t1-impacto-en-trayectoria-del-drone","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/mn-s1eva30\/s1eva2022paoi_t1-impacto-en-trayectoria-del-drone\/","title":{"rendered":"s1Eva2022PAOI_T1 Impacto en trayectoria del drone"},"content":{"rendered":"\n

Ejercicio<\/strong>: 1Eva2022PAOI_T1 Impacto en trayectoria del drone<\/a><\/p>\n\n\n\n

Desarrollo anal\u00edtico<\/h2>\n\n\n\n

a) Realice el planteamiento del problema usando inicialmente las trayectorias en el eje x, donde para el intervalo de operaci\u00f3n del misil antidrone, se observa m\u00e1s de un impacto.<\/p>\n\n\n\n

x1<\/sub>(t) = x2<\/sub>(t)<\/p>\n\n\n\n

f(t) = cos(t) -  sin(0.75 t) =0<\/p>\n\n\n\n

y1<\/sub>(t) = y2<\/sub>(t)<\/p>\n\n\n\n

sin(2 t) =kt<\/p>\n\n\n k = \\frac{sin(2 t)}{t} <\/span>\n\n\n\n

b) Usando el m\u00e9todo de Newton-Raphson<\/strong> encuentre el valor de t<\/strong> en el cual se pretende realizar el impacto al drone. Realice al menos 3 iteraciones de forma anal\u00edtica, use tolerancia de 10-4<\/sup>,<\/p>\n\n\n f(t) = cos(t) - sin(0.75 t) <\/span>\n\n\n f'(t) = - sin(t) - 0.75 cos(0.75 t) <\/span>\n\n\n\n

\"Antidrone<\/figure>\n\n\n\n

Como punto inicial para encontrar la ra\u00edz de f(t) podr\u00eda ser t0<\/sub>=4 para el punto marcado en rojo. Para el m\u00e9todo de Newton-Raphson se tiene que<\/p>\n\n\n t_{i+1} = t_i - \\frac{f(t_i)}{f'(t_i)} <\/span>\n\n\n error = |t_{i+1} - t_i| <\/span>\n\n\n\n

iteraci\u00f3n 1 <\/em><\/strong> t0<\/sub>=4<\/p>\n\n\n t_1 = 4 - \\frac{cos(4) - sin(0.75*4)}{- sin(4) - 0.75 cos(0.75*4)}<\/span>\n\n\n t_1 = 4 - \\frac{-0.7947}{1.4992} = 4.5300 <\/span>\n\n\n tramo = |4.5300-4| = 0.53 <\/span>\n\n\n\n

iteraci\u00f3n 2 <\/em><\/strong> t1<\/sub>=4.53<\/p>\n\n\n t_{2} = 4.53 - \\frac{cos(4.53) - sin(0.75*4.53)}{- sin(4.53) - 0.75 cos(0.75*4.53)} <\/span>\n\n\n t_2 = 4.53 - \\frac{-0.0717}{1.7089} = 4.4880 <\/span>\n\n\n tramo= |4.4880-4.53| = 0.042 <\/span>\n\n\n\n

iteraci\u00f3n 3 <\/em><\/strong> t2<\/sub>=4.4880<\/p>\n\n\n t_{3} = 4.4880 - \\frac{cos(4.4880) - sin(0.75*4.4880)}{ - sin(4.4880) - 0.75 cos(0.75*4.4880)} <\/span>\n\n\n t_3 = 4.4880 - \\frac{-0.0000179}{1.7061} = 4.4879 <\/span>\n\n\n tramo= |4.4879-4.4880| = 0.0001 <\/span>\n\n\n\n

c) Realice el an\u00e1lisis de la convergencia del m\u00e9todo.<\/p>\n\n\n\n

El error disminuye, el m\u00e9todo converge. La raiz se encuentra en t=4.487989505154422<\/p>\n\n\n\n

d) Con el resultado de t<\/strong> anterior, determine el valor de la constante k para la expresi\u00f3n de y2<\/sub>(t) que asegura el impacto contra el drone.<\/p>\n\n\ny_1(t) = y_2(t)<\/span>\n\n\nsin(2 t) =kt<\/span>\n\n\n k = \\frac{sin(2 t)}{t} = \\frac{sin(2*4.487989505154422)}{4.487989505154422} = 0.096714 <\/span>\n\n\n\n

Desarrollo con Algoritmo<\/h2>\n\n\n\n

Resultados<\/p>\n\n\n\n

['xi', 'xnuevo', 'tramo']\n[[4.0000e+00 4.5301e+00 5.3009e-01]\n [4.5301e+00 4.4880e+00 4.2071e-02]\n [4.4880e+00 4.4880e+00 3.0277e-05]]\nraiz en:  4.487989505154422\ncon error de:  3.0276981949128867e-05<\/code><\/pre>\n\n\n\n
Algoritmo en Python<\/p>\n\n\n
\n# M\u00e9todo de Newton-Raphson\nimport numpy as np\n\n# INGRESO\nfx  = lambda t: np.cos(1*t) - np.sin(0.75*t)\ndfx = lambda t: -np.sin(t) - np.cos(0.75*t)*0.75\n\nx0 = 4\ntolera = 0.0001\n\n# PROCEDIMIENTO\ntabla = []\ntramo = abs(2*tolera)\nxi = x0\nwhile (tramo>=tolera):\n    xnuevo = xi - fx(xi)\/dfx(xi)\n    tramo  = abs(xnuevo-xi)\n    tabla.append([xi,xnuevo,tramo])\n    xi = xnuevo\n\n# convierte la lista a un arreglo.\ntabla = np.array(tabla)\nn = len(tabla)\n\n# SALIDA\nprint(['xi', 'xnuevo', 'tramo'])\nnp.set_printoptions(precision = 4)\nprint(tabla)\nprint('raiz en: ', xi)\nprint('con error de: ',tramo)\n<\/pre><\/div>","protected":false},"excerpt":{"rendered":"
Ejercicio: 1Eva2022PAOI_T1 Impacto en trayectoria del drone Desarrollo anal\u00edtico a) Realice el planteamiento del problema usando inicialmente las trayectorias en el eje x, donde para el intervalo de operaci\u00f3n del misil antidrone, se observa m\u00e1s de un impacto. x1(t) = x2(t) f(t) = cos(t) -  sin(0.75 t) =0 y1(t) = y2(t) sin(2 t) =kt b) […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-mn-ejemplo","format":"standard","meta":{"footnotes":""},"categories":[46],"tags":[58,54],"class_list":["post-8286","post","type-post","status-publish","format-standard","hentry","category-mn-s1eva30","tag-ejemplos-python","tag-mnumericos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/8286","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=8286"}],"version-history":[{"count":6,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/8286\/revisions"}],"predecessor-version":[{"id":23837,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/8286\/revisions\/23837"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=8286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=8286"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=8286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}