A resolver problemas

Problema 11. ¿Cuál es la trayectoria más corta?

2. ¿Con qué finalidad se utilizaría?

3. ¿Con cuál se desplazó más?

Problema 2

• ¿Por qué las hormigas siempre terminan llegando a su alimento utilizando la trayectoria más corta?

• Piensen en tres ejemplos más en los que fuera sensato usar dicha línea… como camino para trasladarse.

Colonia de hormigas

• La idea original proviene de la observación de la explotación de los recursos alimentarios entre hormigas, en el que las habilidades cognitivas de las hormigas son individualmente limitadas y en conjunto son capaces de buscar el menor camino existente entre la fuente de comida y su nido o colonia.

• La primera hormiga encuentra la fuente de alimentos (F), a través de cualquier camino (a), entonces retorna a la colonia (N), dejando tras sí un rastro de feromonas;

• Las hormigas indiscriminadamente siguen cuatro caminos posibles, pero el fortalecimiento de la pista hace más atractivo la ruta más corta;

Colonia de hormigas

• Las hormigas toman la ruta más corta y largas porciones de otras rutas empiezan a perder su rastro de feromonas.

• En una serie de experimentos en una colonia de hormigas donde existe la elección de dos rutas diferentes que llevan hasta la fuente de comida, los biólogos observaron que las hormigas tienden a usar la ruta más corta.

• El siguiente modelo explica este comportamiento:

1. Una hormiga (llamada “blitz”) vaguea de manera aleatoria alrededor de la colonia;

2. Si esta encuentra una fuente de comida, ella retorna a la colonia de manera más o menos directa, dejando tras sí un rastro de feromonas;

Colonia de hormigas3. Estas feromonas son atractivas, las hormigas más cercanas se

verán atraídas por estas y seguirán su pista de manera más o menos directa;

4. Regresando a la colonia estas hormigas habrán fortalecido dicha ruta;

5. Si existen dos rutas para que llegan a la misma fuente de alimentos entonces, en una misma cantidad de tiempo dado, la ruta más corta será recorrida por más hormigas que la ruta más larga;

6. La ruta más corta habrá aumentado en cantidad de feromonas y por tanto empezará a ser más atractiva;

7. La ruta más largar irá desapareciendo debido a que las feromonas son volátiles;

8. Finalmente, todas las hormigas habrán determinado y escogido el camino más corto.

Problema 3

• Imagina que planeas hacer un viaje en automóvil desde Mazatlán hasta Veracruz. Según el mapa, la distancia que separa ambos puntos es de aproximadamente 1160 km en línea recta. Pero al consultar una guía de carreteras, te encuentras que la trayectoria más corta por la carretera federal 15 tiene una distancia de 1404 kilómetros.

Problema 3

http://mx.lasdistancias.com/

Problema 3

http://mx.lasdistancias.com/

Preguntas de discusión

• ¿Puedes explicar esta diferencia de kilómetros utilizando los conceptos de distancia, desplazamiento y trayectoria?

• ¿Cuánto tardaría el viaje si la rapidez promedio fuera siempre de 100 km/h?

• ¿Será lo mismo en tiempo y distancia viajar de Mazatlán a Veracruz que de Veracruz a Mazatlán?

• ¿El desplazamiento será el mismo en estos viajes?

Problema 4

• Escuchas por radio que una tormenta eléctrica está viajando con una rapidez de 25 km/h.

• ¿Te deberías preparar para la tormenta?

• ¿Cuál es la diferencia entre rapidez y velocidad?

DESCRIBING MOTION WITH WORDSDistance vs displacement

Speed vs Velocity

http://www.physicsclassroom.com/

1-D Kinematics Lessons a) to d)

http://www.tutorvista.com/physics/

Old word, new concepts

• You have used words and phrases such as going fast, stopped, slowing down, speeding up, and turning provide a sufficient vocabulary for describing the motion of objects.

• We will be expanding upon this vocabulary list with words such as distance, displacement, speed, velocity, and acceleration.

Scalars and vectors

Scalars vs vectors

• The mathematical quantities that are used to describe the motion of objects can be divided into two categories.

• The quantity is either a vector or a scalar.

• These two categories can be distinguished from one another by their distinct definitions:

Scalars

• Scalars are quantities that are fully described by a magnitude (or numerical value) alone.

Vectors

• Vectors are quantities that are fully described by both a magnitude and a direction.

Common directions:

• Forward• Backward• Northward• Southward• Eastward• Westward

Examples

Exercise 01

• Consider the following quantities listed below. Categorize each quantity as being either a vector or a scalar.

a) 5 metersb) 30 m/sec, East c) 5 miles, North d) 20 degrees Celsius e) 256 bytes f) 4000 Calories

Distance vs displacement

Distance and Displacement

• Distance and displacement are two quantities that may seem to mean the same thing yet have distinctly different definitions and meanings.

Distance

• Distance is a scalar quantity that refers to "how much ground an object has covered" during its motion.

• Distance can be defined as how far two objects are. Distance in physics may refer to physical length.

• Distance is the scalar path between two locations measured along path connecting them.

Example 01

• In the right figure you can see a rat and cheese.  If the rat want the cheese it must follow the blue path. And the length of blue path is the distance. Or it can be said in another way. How far the rat is from cheese. If you ask this question your self the result you get is nothing but the distance.

Displacement

• Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.

• Displacement can be defined as shortest distance between the starting point and end point.

• Displacement can be defined as imaginary straight path which is different from the actual path.

Example 01

• In the left figure the green path present between the rat and cheese is the displacement because it is the shortest path.

• So by this we can conclude displacement is a vector measure of the interval between two locations measured along the shortest path connecting them

Example 02

Example 03

• A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North.

• Even though the physics teacher has walked a total distance of 12 meters, her displacement is 0 meters.

• During the course of her motion, she has "covered 12 meters of ground" (distance =12 m).

• Yet when she is finished walking, she is not "out of place" - i.e., there is no displacement for her motion (displacement = 0 m).

Example 04• You ride your bicycle from

Manhattan to New Jersey.• Getting there is a three step

process.1. Follow the Hudson River 8.2 km

upriver.2. Cross using the George

Washington Bridge (1.8 km between anchorages).

3. Reverse direction and head down river for 4.5 km.

• The distance traveled is a reasonable 14 km, but the resultant displacement is a mere 2.7 km north. The end of this journey is actually visible from the start. Maybe I should buy a canoe.

Example 05• Use the diagram to determine the resulting displacement

and the distance traveled by the skier during these three minutes.

• The skier covers a distance of (180 m+140 m+100 m) = 420 m and has a displacement of 140 m, rightward.

Exercise 01

• The distance traveled by an object is the total length that is traveled by that object.

• Displacement of an object from a point of reference, O is the shortest distance of the object from point O in a specific direction.

• What is the distance travelled and the displacement?

Answer 01

• Distance travelled = 200mDisplacement = 120 m, in the direction of Northeast

• Distance is a scalar quantity,

• Displacement is a vector quantity

Exercise 02

• What is the displacement of the cross-country team if they begin at the school, run 10 miles and finish back at the school?

Answer 02

• The displacement of the runners is 0 miles.

• While they have covered a distance of 10 miles, they are not "out of place" or displaced.

• They finish where they started. Round-trip motions always have a displacement of 0.

Exercise 03

• What is the distance and the displacement of the race car drivers in the Indy 500?

Answer 03

• The displacement of the cars is somewhere near 0 miles since they virtually finish where they started.

• Yet the successful cars have covered a distance of 500 miles.

Exercise 04

• What’s the distance?• What is the displacement?

Answer 04

• In the last figure we can see earth moving around the sun. Let the earth begin its journey around the sun from point A. After one year the again it will reach the point A.

• Now the path which the earth followed is elliptical path. So the distance travelled is nothing but the circumference of that elliptical path.

How far does the Earth travel in one year?

• In terms of distance, quite far (the circumference of the earth's orbit is nearly one trillion meters), but in terms of displacement, not far at all (zero, actually).

• At the end of a year's time the earth is right back where it started from. It hasn't gone anywhere.

• Since both the starting point and the end point are same it is zero. Hard to believe but it is true. By this you can understand the difference between distance and displacement.

Exercise 05

Describe distance travelled and displacement.

Answer 05

• IN THE FIGURE LET THE STARTING POINT OF THE EARTH BE A AND POINT BE D. NOW

• DISTANCE= • CIRCULAR LENGTH BETWEEN A AND B +

CIRCULAR LENGTH BETWEEN B AND C + CIRCULAR LENGTH BETWEEN C AND D.

• DISPLACEMENT = LENGTH OF BLACK LINE WHICH YOU CAN SEE FROM THE FIGURE.

• THIS IS ALL ABOUT DIFFERENCE BETWEEN DISTANCE AND DISPLACEMENT.

• HOPE YOU ENJOYED IT !

Speed vs Velocity

Reactivate

• What do you think when you hear the word “Speed”?

• Describe a sport or activity in which speed is important?

• What examples are there of velocity?

• Do these concepts mean the same?

Speed

• The motion of an airplane is fast. (260 m / second)

• The motion of a snail is slow. (1 mm / second)

• By using these words, your are describing the object’s speed.

• The speed or the airplane is much greater that the speed of the snail because the airplane travels mucho farther than the snail in the same amount of time.

Gary the snail

Speed (Scalar quantity)

• Refers to "how fast an object is moving“

• Speed is the rate of change in distance with respect to time.

• A fast-moving object has a high speed while a slow-moving object has a low speed.

• An object with no movement at all has a zero speed.

So, What is speed?

• The speed of an object is the distance the object moves per unit of time.

• To calculate the speed of an object, divide the distance the object travels by the amount of time it takes to travel that distance. (m/s = meters per second)

What is the relationship between speed, traveled distance and time?

Calculating Speed

• The motorcyclist has traveled a total distance of 24 Kilometers in 3 hours. What has his average Speed?

Exercise 01

• While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8 hours.

• What was her average speed?

Exercise 02

• You are driving to Mexico to visit your Grandmas to eat her scrumpcious snickerdoodle cookies. In 5 hours you drove 1000 meters to stop for milk. You still had 940 more meters to go.

• If you got to grandmas house in a total of 8 hours what is your average velocity between your milk rest and grandmas.

Velocity (vector quantity)

• Refers to "the rate at which an object changes its position“.

• Velocity is the rate of change in displacement with respect to time.

Example

• You hear that a thunderstorm is travelling at a speed of 25 km/h. Should you prepare for the storm?– Can you know the velocity of the storm?– What do you need?

Exercise 01

• The physics teacher problem

• ¿What was the distance travelled?

• If he entire motion lasted 24 seconds ¿What was the average speed?

• ¿What is the velocity of the teacher?

Air traffic controllers & stunt pilots

Exercise 02

• A car travels 90. meters due north in 15 sec. • Then the car turns around and travels 40.

meters due south in 5.0 seconds. • What is the magnitude of the average velocity of

the car during this 20 - second interval? • A)2.5 m/s• B)5.0 m/s• C)6.5 m/s• D)7.5 m/s

Exercise 03

• The position of an object is +35 meters at 2.0 seconds and is +87 meters at 15 seconds.

• Calculate the average velocity of the object.

Remember

• If you know the distance an object travels in a certain amount of time, you can calculate the speed of the object.

• When you know both the speed and the direction of an object´s motion, you know the velocity of the object.