MARCO TEORICO

Qu es una Curva de persecucin?

Una curva de bsqueda es el camino que ocupa un objeto cuando persigue otro objeto.

Por ejemplo:

Si un zorro persigue a un conejo, la curva de bsqueda sera la ruta de la zorra toma mientras se ejecuta despus del conejo.

DATOS

El perseguidor (zorro) siempre la cabeza apunta directamente hacia la posicin del perseguido (Conejo).

La velocidad del perseguido es proporcional a la velocidad del perseguidor

VECTOR UNITARIO

ENUNCIADO DEL PROBLEMA

Analizar el movimiento que realiza un zorro que inicia la persecucin de un conejo que describe una curva. El conejo corre con una celeridad constante de 6 m/s siguiendo una trayectoria circular de 12 m de radio t el zorro corre con celeridad constate de 7 m/s. El camino que sigue esta siempre dirigido hacia la posicin del conejo en cada instante.

SOLUCIN

1.- La curva que describe el perseguido (conejo) est definida por la ecuacin vectorial:

p y q son funciones del tiempo (t).

2.- La curva que describe el perseguidor (zorro) est definida por la ecuacin vectorial:

3.-el vector unitario de la velocidad () es igual al vector unitario tangente de F que representa el vector unitario que une la posicin del zorro hacia la posicin del conejo.

Adems por condicin: el modulo velocidad del zorro () es directamente proporcional al mdulo velocidad del conejo:

Entonces:

4.-El vector unitario que une la posicin del zorro hacia la posicin del conejo tambin puede ser representado por el vector unitario de la diferencia de:

5.- y representan el vector unitario que une la posicin del zorro hacia la posicin del conejo.

Por lo tanto

6.- De las ecuaciones (1) y (2) se obtiene:

7.- Reemplazando las ecuaciones obtenidas en la ecuacin 7:

8.-Igualando las componentes en las direcciones i y j:

Multiplicado por :

9.-dichas ecuaciones se pueden representar por:

Las ecuaciones obtenidas definen las componentes x y y de la curva que describe el zorro en persecucin del conejo que se describe una curva y cuyas velocidades son directamente proporcionales.

APLICANDO LAS ECUACIONES POR ITERACIONES (mtodos numricos)

Dado la posicin de conejo segn condiciones del problema que se mueve alrededor de un circulo con coordenadas X, Y (P (), Q ())

p ()= 12*c os ()q ()= 12*sin ()

Para realizar las iteraciones tomamos tiempos pequeos de cada 0.005

=0

=0

Calculamos x1, y1:

Luego con los x1 y y1 obtenidos hallamos x2, y2 y as sucesivamente.

PRIMER EJEMPLO

DESARRROLLO CON EL Dev-C++

DESARRROLLO CON EL Excel

t(seg)

x(m)

y(m)

x2+y2

r(m)

p(m)

q(m)

1

0

0

0

0

0

12

0

2

0.005

0.035

0

0.001225

0.035

11.999963

0.03

3

0.01

0.07

0.000088

0.00490001

0.07000006

11.99985

0.06

4

0.015

0.104999

0.000264

0.01102486

0.10499933

11.999663

0.089999

5

0.02

0.139998

0.000528

0.01959972

0.139999

11.9994

0.119998

6

0.025

0.174997

0.00088

0.03062472

0.17499921

11.999063

0.149996

7

0.03

0.209994

0.001321

0.04409923

0.20999815

11.99865

0.179993

8

0.035

0.24499

0.001852

0.06002353

0.244997

11.998163

0.209989

9

0.04

0.279984

0.002472

0.07839715

0.27999491

11.9976

0.239984

10

0.045

0.314977

0.003181

0.09922063

0.31499306

11.996963

0.269977

11

0.05

0.349968

0.00398

0.12249344

0.34999063

11.99625

0.299969

12

0.055

0.384957

0.004869

0.1482156

0.38498779

11.995463

0.329958

13

0.06

0.419943

0.005849

0.17638633

0.41998373

11.9946

0.359946

14

0.065

0.454927

0.006919

0.20700645

0.45497961

11.993663

0.389931

15

0.07

0.489907

0.00808

0.24007416

0.48997363

11.992651

0.419914

16

0.075

0.524885

0.009333

0.27559137

0.52496797

11.991563

0.449895

17

0.08

0.559859

0.010676

0.31355608

0.55996078

11.990401

0.479872

18

0.085

0.59483

0.012112

0.35396943

0.5949533

11.989164

0.509846

19

0.09

0.629796

0.013639

0.39682902

0.62994367

11.987852

0.539818

20

0.095

0.664759

0.015259

0.44213737

0.66493411

11.986465

0.569786

21

0.1

0.699717

0.016971

0.48989189

0.69992278

11.985003

0.59975

22

0.105

0.73467

0.018776

0.54009255

0.73490989

11.983466

0.629711

23

0.11

0.769619

0.020674

0.59274082

0.76989663

11.981855

0.659667

24

0.115

0.804562

0.022666

0.64783376

0.80488121

11.980168

0.68962

25

0.12

0.8395

0.024751

0.70537286

0.83986479

11.978406

0.719568

DESRROLLO CON EL MATLAB

SEGUNDO EJEMPLO

DESARRROLLO CON EL Dev-C++

DESARRROLLO CON EL Excel

t(seg)

x(m)

y(m)

x2+y2

r(m)

p(m)

q(m)

1

0

24

0

576

24

12

0

2

0.01

23.93

0

572.6449

23.93

11.99985

0.06

3

0.02

23.860001

0.000352

569.299648

23.860001

11.9994

0.119998

4

0.03

23.790005

0.001058

565.964339

23.790005

11.99865

0.179993

5

0.04

23.720013

0.00212

562.639021

23.7200131

11.9976

0.239984

6

0.05

23.650027

0.00354

559.32379

23.6500273

11.99625

0.299969

7

0.06

23.58005

0.00532

556.018786

23.5800506

11.9946

0.359946

8

0.07

23.510083

0.007462

552.724058

23.5100842

11.992651

0.419914

9

0.08

23.440128

0.009967

549.4397

23.4401301

11.990401

0.479872

10

0.09

23.370187

0.012838

546.165805

23.3701905

11.987852

0.539818

11

0.1

23.300262

0.016075

542.902468

23.3002675

11.985003

0.59975

12

0.11

23.230355

0.019681

539.649781

23.2303633

11.981855

0.659667

13

0.12

23.160468

0.023657

536.407838

23.1604801

11.978406

0.719568

14

0.13

23.090603

0.028005

533.176731

23.09062

11.974659

0.779451

15

0.14

23.020762

0.032727

529.956554

23.0207853

11.970612

0.839314

16

0.15

22.950948

0.037822

526.747445

22.9509792

11.966266

0.899156

17

0.16

22.881163

0.043295

523.549495

22.881204

11.96162

0.958976

18

0.17

22.811407

0.049144

520.362704

22.8114599

11.956676

1.018772

19

0.18

22.741685

0.055372

517.187303

22.7417524

11.951433

1.078543

20

0.19

22.671998

0.06198

514.023335

22.6720827

11.945891

1.138286

21

0.2

22.602348

0.068969

510.870892

22.6024532

11.94005

1.198001

22

0.21

22.532737

0.07634

507.730065

22.5328663

11.933911

1.257686

23

0.22

22.463168

0.084094

504.600988

22.4633254

11.927473

1.31734

24

0.23

22.393643

0.092233

501.483754

22.3938329

11.920737

1.37696

25

0.24

22.324163

0.100756

498.378405

22.3243904

11.913704

1.436546

DESRROLLO CON EL MATLAB

t vs y

y= -0.0703t6 + 0.923t5 - 4.4481t4 + 8.7977t3 - 4.9074t2 + 1.9557t- 0.1309

05.0000000000000001E-30.011.4999999999999999E-20.022.5000000000000001E-20.033.5000000000000003E-20.044.4999999999999998E-20.055.5E-20.066.5000000000000002E-27.0000000000000007E-27.4999999999999997E-20.088.5000000000000006E-20.099.5000000000000001E-20.10.1050.110.1150.120.1250.130.135000000000000010.140000000000000010.144999999999999990.150.1550.160.165000000000000010.170.174999999999999990.180.1850.190.195000000000000010.20.204999999999999990.210.2150.220.225000000000000010.230.234999999999999990.240.2450.250.2550.260.265000000000000010.270.275000000000000020.280000000000000030.284999999999999980.289999999999999980.294999999999999980.30.304999999999999990.310.3150.320.325000000000000010.330.335000000000000020.340.344999999999999970.350.354999999999999980.360.364999999999999990.370.3750.380.385000000000000010.390.395000000000000020.40.405000000000000030.410.414999999999999980.420.424999999999999990.430.4350.440.445000000000000010.450.455000000000000020.460.465000000000000020.470.474999999999999980.480.484999999999999990.490.4950.50.5050.510.515000000000000010.520.525000000000000020.530.535000000000000030.540.545000000000000040.550000000000000040.555000000000000050.560000000000000050.564999999999999950.569999999999999950.574999999999999960.579999999999999960.584999999999999960.590.594999999999999970.60.604999999999999980.610.614999999999999990.620.6250.630.635000000000000010.640.645000000000000020.650.655000000000000030.660.665000000000000040.670.675000000000000040.680.685000000000000050.690.694999999999999950.70.704999999999999960.710.714999999999999970.720.724999999999999980.730.734999999999999990.740.7450.750.7550.760.765000000000000010.770.775000000000000020.780.785000000000000030.790.795000000000000040.80.805000000000000050.810.814999999999999950.820.824999999999999960.830.834999999999999960.840.844999999999999970.850.854999999999999980.860.864999999999999990.870.8750.880.885000000000000010.890.895000000000000020.90.905000000000000030.910.915000000000000040.920.925000000000000040.930.935000000000000050.940.944999999999999950.950.954999999999999960.960.964999999999999970.970.974999999999999980.980.984999999999999990.990.99511.00499999999999991.011.01499999999999991.021.02499999999999991.031.03499999999999991.041.04499999999999991.051.05499999999999991.061.06499999999999991.071.0751.081.0851.09000000000000011.0951.10000000000000011.1051.11000000000000011.1151.12000000000000011.1251.12999999999999991.1351.13999999999999991.1451.14999999999999991.1551.15999999999999991.1651.171.1751.181.18500000000000011.191.19500000000000011.21.20500000000000011.211.21500000000000011.221.22500000000000011.231.23500000000000011.241.24500000000000011.251.25499999999999991.261.26499999999999991.271.27499999999999991.281.28499999999999991.291.29499999999999991.31.30499999999999991.311.31499999999999991.321.3251.331.3351.341.3451.351.3551.361.3651.371.3751.381.3851.391.3951.41.4051.411.4151.421.4251.431.43500000000000011.441.44500000000000011.451.45500000000000011.461.46500000000000011.471.47500000000000011.481.48500000000000011.491.49500000000000011.51.50499999999999991.511.51499999999999991.521.52499999999999991.531.53499999999999991.541.54499999999999991.551.55499999999999991.561.56499999999999991.571.5751.581.5851.591.5951.61.6051.611.6151.621.6251.631.6351.641.6451.651.6551.661.6651.671.6751.681.68500000000000011.691.69500000000000011.71.70500000000000011.711.71500000000000011.721.72500000000000011.731.73500000000000011.741.74500000000000011.751.75499999999999991.761.76499999999999991.771.77499999999999991.781.78499999999999991.791.79499999999999991.81.80499999999999991.811.81499999999999991.821.8251.831.8351.841.8451.851.8551.861.8651.871.8751.881.8851.891.8951.91.9051.911.9151.921.9251.931.93500000000000011.941.94500000000000011.951.95500000000000011.961.96500000000000011.971.97500000000000011.981.98500000000000011.991.995000000000000122.00499999999999992.00999999999999982.01500000000000012.022.02499999999999992.02999999999999982.03500000000000012.042.04499999999999992.04999999999999982.05500000000000022.062.06499999999999992.06999999999999982.07500000000000022.082.0852.092.09500000000000022.12.1052.112.11500000000000022.122.1252.132.13499999999999982.142.1452.152.15499999999999982.162.1652.172.17499999999999982.18000000000000022.18500000000000012.192.19499999999999982.20000000000000022.20500000000000012.212.21499999999999992.22000000000000022.22500000000000012.232.23499999999999992.24000000000000022.24500000000000012.252.25499999999999992.25999999999999982.26500000000000012.272.27499999999999992.27999999999999982.28500000000000012.292.29499999999999992.29999999999999982.30500000000000022.312.31499999999999992.31999999999999982.32500000000000022.332.3352.342.34500000000000022.352.3552.362.36500000000000022.372.3752.382.38499999999999982.392.3952.42.40499999999999982.412.4152.422.42499999999999982.43000000000000022.43500000000000012.442.44499999999999982.45000000000000022.45500000000000012.462.46499999999999992.47000000000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t(seg)

y(m)

t vs x

x= -0.0639t6 + 0.6058t5 - 1.6232t4 + 0.0933t3 + 1.233t2 + 6.239t+ 0.0756

05.0000000000000001E-30.011.4999999999999999E-20.022.5000000000000001E-20.033.5000000000000003E-20.044.4999999999999998E-20.055.5E-20.066.5000000000000002E-27.0000000000000007E-27.4999999999999997E-20.088.5000000000000006E-20.099.5000000000000001E-20.10.1050.110.1150.120.1250.130.135000000000000010.140000000000000010.144999999999999990.150.1550.160.165000000000000010.170.174999999999999990.180.1850.190.195000000000000010.20.204999999999999990.210.2150.220.225000000000000010.230.234999999999999990.240.2450.250.2550.260.265000000000000010.270.275000000000000020.280000000000000030.284999999999999980.289999999999999980.294999999999999980.30.304999999999999990.310.3150.320.325000000000000010.330.335000000000000020.340.344999999999999970.350.354999999999999980.360.364999999999999990.370.3750.380.385000000000000010.390.395000000000000020.40.405000000000000030.410.414999999999999980.420.424999999999999990.430.4350.440.445000000000000010.450.455000000000000020.460.465000000000000020.470.474999999999999980.480.484999999999999990.490.4950.50.5050.510.515000000000000010.520.525000000000000020.530.535000000000000030.540.545000000000000040.550000000000000040.555000000000000050.560000000000000050.564999999999999950.569999999999999950.574999999999999960.579999999999999960.584999999999999960.590.594999999999999970.60.604999999999999980.610.614999999999999990.620.6250.630.635000000000000010.640.645000000000000020.650.655000000000000030.660.665000000000000040.670.675000000000000040.680.685000000000000050.690.694999999999999950.70.704999999999999960.710.714999999999999970.720.724999999999999980.730.734999999999999990.740.7450.750.7550.760.765000000000000010.770.775000000000000020.780.785000000000000030.790.795000000000000040.80.805000000000000050.810.814999999999999950.820.824999999999999960.830.834999999999999960.840.844999999999999970.850.854999999999999980.860.864999999999999990.870.8750.880.885000000000000010.890.895000000000000020.90.905000000000000030.910.915000000000000040.920.925000000000000040.930.935000000000000050.940.944999999999999950.950.954999999999999960.960.964999999999999970.970.974999999999999980.980.984999999999999990.990.99511.00499999999999991.011.01499999999999991.021.02499999999999991.031.03499999999999991.041.04499999999999991.051.05499999999999991.061.06499999999999991.071.0751.081.0851.09000000000000011.0951.10000000000000011.1051.11000000000000011.1151.12000000000000011.1251.12999999999999991.1351.13999999999999991.1451.14999999999999991.1551.15999999999999991.1651.171.1751.181.18500000000000011.191.19500000000000011.21.20500000000000011.211.21500000000000011.221.22500000000000011.231.23500000000000011.241.24500000000000011.251.25499999999999991.261.26499999999999991.271.27499999999999991.281.28499999999999991.291.29499999999999991.31.30499999999999991.311.31499999999999991.321.3251.331.3351.341.3451.351.3551.361.3651.371.3751.381.3851.391.3951.41.4051.411.4151.421.4251.431.43500000000000011.441.44500000000000011.451.45500000000000011.461.46500000000000011.471.47500000000000011.481.48500000000000011.491.49500000000000011.51.50499999999999991.511.51499999999999991.521.52499999999999991.531.53499999999999991.541.54499999999999991.551.55499999999999991.561.56499999999999991.571.5751.581.5851.591.5951.61.6051.611.6151.621.6251.631.6351.641.6451.651.6551.661.6651.671.6751.681.68500000000000011.691.69500000000000011.71.70500000000000011.711.71500000000000011.721.72500000000000011.731.73500000000000011.741.74500000000000011.751.75499999999999991.761.76499999999999991.771.77499999999999991.781.78499999999999991.791.79499999999999991.81.80499999999999991.811.81499999999999991.821.8251.831.8351.841.8451.851.8551.861.8651.871.8751.881.8851.891.8951.91.9051.911.9151.921.9251.931.93500000000000011.941.94500000000000011.951.95500000000000011.961.96500000000000011.971.97500000000000011.981.98500000000000011.991.995000000000000122.00499999999999992.00999999999999982.01500000000000012.022.02499999999999992.02999999999999982.03500000000000012.042.04499999999999992.04999999999999982.05500000000000022.062.06499999999999992.06999999999999982.07500000000000022.082.0852.092.09500000000000022.12.1052.112.11500000000000022.122.1252.132.13499999999999982.142.1452.152.15499999999999982.162.1652.172.17499999999999982.18000000000000022.18500000000000012.192.19499999999999982.20000000000000022.20500000000000012.212.21499999999999992.22000000000000022.22500000000000012.232.23499999999999992.24000000000000022.24500000000000012.252.25499999999999992.25999999999999982.26500000000000012.272.27499999999999992.27999999999999982.28500000000000012.292.29499999999999992.29999999999999982.30500000000000022.312.31499999999999992.31999999999999982.32500000000000022.332.3352.342.34500000000000022.352.3552.362.36500000000000022.372.3752.382.38499999999999982.392.3952.42.40499999999999982.412.4152.422.42499999999999982.43000000000000022.43500000000000012.442.44499999999999982.45000000000000022.45500000000000012.462.46499999999999992.47000000000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t(seg)

x(m)

x vs y

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