Two Dimensional Motion

Transcription

1 Two Dimensional Motion Chapters 3 & 4 Geometric/Graphical Representation VECTORS 1

2 Vectors For 1-D vectors size is designated b a number and direction is designated b a sign. E.g. +3m/s, -9.8m/s For -D vectors size is designated b a number and direction is designated b an angle. E.g. 3 65, Alternativel, -D vectors can be designated b a pair of signed numbers (components). Graphical Representation Vectors can be represented graphicall b directed ras (i.e. arrows). The length of the arrow is proportional to the magnitude of the vector and the direction the arrow points is the direction the vector points.

3 Adding Vectors Geometricall If ou move from point A to point B, and then from point B to point C, ou have moved two displacements: AB and BC. Your net displacement is given b the vector sum, or resultant, AC. We can represent the relation among the three vectors b the following vector equation: s a b Adding Vectors Geometricall With a ruler and protractor: 1. Sketch vector a to some convenient scale at the proper angle. Sketch vector b to the same scale, with its tail at the head vector a, again at the proper angle 3. The vector sum s is the vector that extends from the tail of a to the head of b. 3

4 Vector Addition: Commutative Law The order of addition of two or more vectors does not matter. Adding a to b gives the same result as adding b to a. a b b a Vector Addition: Associative Law When there are more than two vectors, we can group them in an order as we add them. We can add a and b and then add c to the resultant, or we can add b and c and then and a. We get the same result either wa. ( a b) c a ( b c) 4

5 Negative Vectors The vector b is a vector with the same magnitude as b, but the opposite direction. Adding the two vectors would ield zero. b ( b) Vector Subtraction Adding b has the same effect as subtraction b We use this propert to define the difference between two vectors. d a b a ( b) 5

6 Checkpoint 1 The magnitudes of displacements a and b are 3 m and 4 m respectivel, and c =a + b. Considering various orientations of a and b, what is: a) the maximum possible magnitude c? b) the minimum possible magnitude? Sample 1 In orienteering class, ou have the goal of moving as far (straight-line distance) from base camp as possible b making three straight-line moves. You ma use the following displacements in an order: a =. km due east b =. km 3 north of east c = 1. km due west Alternativel, ou ma substitute either b for b or c or c. What is the greatest distance ou can be from base cap at the end of the third displacement? 6

7 Analtic/Numerical Representation VECTORS Vector Components Vector components are projections of a vector onto a perpendicular axes. The components, a x and a, of vector a are shown. The process of finding the components of a vector is called resolving the vector. 7

8 Vector Components When the components of a vector are moved to form a head-to-toe chain, we see that a right triangle is formed whose hpotenuse is the magnitude of the vector. Vector Components Trigonometr defines three basic relationships for right triangles: sin(q) = o/h cos(q) = a/h tan(q) = o/a SOH CAH TOA hpotenuse q adjacent opposite 8

9 Checkpoint In the figure, which of the indicated methods of combining the x and components of vector a are proper to determine that vector? Sample A small airplane leaves an airport on an overcast da and is later sighted 15 km awa, in a direction making an angle of east of due north. How far east and north is the airplane from the airport when sighted? 9

10 Sample 3 A velocit vector is 5 m/s at 53 south of east. What are the sizes and signs of the x- and -components of this velocit? Component Notation The components of a vector can be used to describe a vector just as accuratel as its magnitude and angle. This is known as component notation. Magnitude-angle notation: a = 1 3 Component notation: a = <86.7m, 5m> 1

11 Combining Components Because the components of a vector form a right triangle with the vector itself, we can use the following geometric functions to combine the components of a vector, or change its components into a magnitude and an angle. a a tanq x a a a x Sample 4 For two decades, spelunking teams sought a connection between the Flint Ridge cave sstem and the Mammoth Cave, which are in Kentuck. When the connection was finall discovered, the combined sstem was declared the world s longest cave (more than km long). The team that found the connection had to crawl, climb, and squirm through countless passages, traveling a net.6 km westward, 3.9 km southward, and 5 m upward. What was their displacement from start to finish? 11

12 Angles Degrees and Radians Angles that are measured relative to the positive direction of the x axis are positive in the counterclockwise direction and negative if measured clockwise. For example 1 and -15 are the same angle. Angles ma be measured in degrees or radians (rad). To relate the two measures, recall that a full circle is 36 and π rad. Ex: 4 =.7 rad Inverse Trig Functions When the inverse trig functions are taken on a calculator, ou must consider the reasonableness of the answer ou get, because there is usuall another possible answer that the calculator does not give. The range of operation for a calculator in taking each inverse trig function is indicated on the graphs. The correct answer is the one that seems more reasonable for the given situation. 1

13 Unit Vectors A unit vector is a vector that has a magnitude of exactl 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point that is, to specif a direction. The unit vectors in the positive directions of the x,, and z axes are labeled iˆ, ˆ, j and kˆ. The arrangement of axis shown below is known as the right-handed coordinate sstem and we will be using it exclusivel. Unit-Vector Notation Unit vectors can be used to express other vectors via the scalar components of those vectors. This is known as unit-vector notation. Magnitude-angle notation: a = 1 3 Component notation: a = <86.7 m, 5 m> Unit-vector notation: a = (86.7 m)î + (5 m)ĵ 13

14 Adding (and Subtracting) Vectors b Components Vectors can be added and subtracted analticall using their components. 1. Resolve the vectors into their scalar components. Combine the scalar components, axis b axis, to get the components of the sum 3. Combine the components and express in magnitude-angle A notation, component notation, or unit-vector notation. A iˆ A x ˆj B B iˆ B x ˆj C C iˆ C A B ( Ax Bx)ˆ i ( A B ) ˆj A B C ( A B C )ˆ i ( A B C x x x x ) ˆj ˆj Checkpoint 3 (a) In the figure, what are the signs of the x components of d 1 and d? (b) What are the signs of the components d 1 and d? (c) What are the signs of the x and components of d 1 + d? 14

15 Sample 5 The shows the following three vectors: a (4.m)ˆ i (1.5m) ˆj b ( 1.6m)ˆ i (.9m) ˆj c ( 3.7m) ˆj What is their vector sum r? Sample 6 A fellow camper is to walk awa from ou in a straight line (vector A), turn, walk in a second straight line (vector B) and then stop. How far must ou walk in a straight line (vector C) to reach her? The three vectors are related b C = A + B. A has a magnitude of. m and is directed at an angle of -47 from the positive direction of an x axis. B has a magnitude of 17. m and is directed counterclockwise from the positive direction of the x axis b angle φ. C is in the positive direction of the x axis. What is the magnitude of C? 15

16 Sample 7 The desert ant Cataglphis fortis lives in the plains of the Sahara desert. When one of the ants forages for food, it travels from its home nest along a haphazard search path. The ant ma travel more than 5 m along such a complicated path over flat, featureless sand that contains no landmarks. Yet, when the ant decides to return home, it turns and then runs directl home. How does the ant know the wa home with no guiding clues on the desert plain? According to experiments, the desert ant keeps track of its movements along a mental coordinate sstem. When it wants to return to its home nest, it effectivel sums its displacements along the axes of the sstem to calculate a vector that points directl home. As an example of the calculation, let s consider an ant making five runs of 6. cm each on an x coordinate sstem, in the direction shown in the figure on the next slide, starting from home. At the end of the fifth run, what are the magnitude and angle of the ant s net displacement vector d net? And what are those of the homeward vector d home that expends from the ant s final position back to home. 16

17 MULTIPLYING VECTORS Multipling a Vector b a Scalar Magnitude: ab a b Direction: Same as original vector if scalar is positive Opposite original vector is scalar is negative 17

18 Multipling a Vector b a Vector Two was: One results in a scalar (Scalar Product) One results in a new vector (Vector Product) Scalar Product (Dot Product) a b abcos A dot product can be regarded as the product of two quantities: (1) the magnitude of one of the vectors and () the scalar component of the second vector along the direction of the first vector. 18

19 Scalar Product (Dot Product) If the angle φ between the two vectors is, the component of one vector along the other is maximum, and so also is the dot product of the vectors. If φ is 9, the component of one vector along the other is zero, and so is the dot product. a b ( a iˆ a ˆj a kˆ) ( b iˆ b ˆj b kˆ) x a b a b x x z a b x a b z z z Checkpoint 4 Vectors C and D have magnitudes of 3 units and 4 units, respectivel. What is the angle between the directions of C and D if C D equals (a) zero (b) 1 units (c) -1 units 19

20 Sample 8 a 3.ˆ i 4. ˆj b.ˆ i 3.kˆ What is the angle φ between and? Vector Product (Cross Product) a b absinq The vector product produces a new vector that is perpendicular to both of the original vectors, i.e. perpendicular to the plan that contains both of the original vectors. If a and b are parallel or antiparallel, a x b =. The magnitude of a x b, is maximum when a and b are perpendicular to each other.

21 Right-hand Rule The direction of the resultant vector of a cross product can be found using the right-hand rule. 1. Place the two vectors ou are crossing tail to tail without altering their orientations. Place our right hand at the origin of the two vectors such that ou fingers would sweep from the first vector into the second through the smallest angle between them. 3. Your outstretched thumb points in the direction of the vector product. Checkpoint 5 Vectors C and D have magnitudes of 3 units and 4 units, respectivel. What is the angle between the directions of C and D if the magnitude of the vector product C x D is (a) zero (b) 1 units (c) -1 units 1

22 Sample 9 Vector a lies in the x plane, has a magnitude of 18 units and points in a direction 5 from the +x direction. Also, vector b has a magnitude of 1 units and points in the +z direction. What is the vector product c = a x b? Sample 1 If a iˆ ˆj and b iˆ 3kˆ 3 4, what is c a b?

23 Common Errors with Cross Products 1. Failure to arrange vectors tail to tail is tempting when an illustration presents them head to tail. Redraw one vector to the proper arrangement.. Failing to us the right hand in appling the right-hand rule is eas when the right hand is occupied with a calculator or pencil. 3. Failure to sweep the first vector of the product into the second vector can occur when the orientations of the vectors require an awkward twisting of our hand to appl the right-hand rule. Sometimes that happens when ou tr to make the sweep mentall rather than actuall using our hand. 4. Failure to work with the right-handed coordinate sstem results when ou forget how to draw such a sstem. Practice drawing other prespectives. MOTION QUANTITIES IN VECTOR FORM 3

24 Position The general wa of locating a particle is with a position vector r, which is a vector that extends from a reference point (or origin) to the particle. r xiˆ j ˆ zkˆ Displacement If a particles position vector changes during a certain time interval, then the particles displacement vector Δr is given b: r r r1 r ( x x )ˆ i ( ) ˆj ( z z ) kˆ r xiˆ j ˆ zkˆ 4

25 Sample 11 The position vector for a particle initiall is r ( 3.m)ˆ i (.m) ˆj And then later is r m i m) ˆj m kˆ (9. )ˆ (. (8. ). What is the particle s displacement Δr from r 1 to r? 1 (5.m) kˆ Sample 1 A rabbit runs across a parking lot on which a set of coordinate axes has, strangel enough, been drawn. The coordinates of the rabbit s position as functions of time are given b: (a) At t = 15 s, what is the rabbit s position vector r in unit-vector notation and in magnitude angle notation? (b) Graph the rabbits path for t = to t = 5s. 5

26 Average Velocit Average velocit = displacement/time interval v avg r vavg t x iˆ ˆj t t z t kˆ v Instantaneous Velocit dr v dt dx d dz iˆ ˆj kˆ dt dt dt v v iˆ v ˆj v kˆ x z The direction of the instantaneous velocit v of a particle is alwas tangent to the particle s path at the particle s position 6

27 Checkpoint 7 The figure shows a circular path taken b a particle. If the instantaneous velocit of the particle is v ( m/ s)ˆ i (m/ s) ˆj, through which quadrant is the particle moving at that instant if it is traveling (a) clockwise and (b) counterclockwise around the circle? For both cases, draw v on the figure. Sample 13 For the rabbit in Sample 1, find the velocit v at time t= 15s, in unitvector notation and in magnitude-angle notation. 7

28 Average Acceleration Average acceleration = change in velocit/time interval a avg v v t 1 v t Instantaneous Acceleration dv a dt dv dv x dv a iˆ ˆj dt dt dt a a iˆ a ˆj a kˆ x z z kˆ Caution: When an acceleration vector is drawn it does not extend from one position to another. Rather, it shows the direction of acceleration for a particle located at its tail, and its length (representing the acceleration magnitude) can be drawn to an scale. 8

29 Checkpoint 8 Here are four descriptions of the position (in meters) of a puck as it moves in an x plane: x 3t 4t (1) and 3 () x 3t 4t and (3) r t iˆ (4t 3) ˆj 3 (4) r (4t t)ˆ i 3 ˆj 6t 5t 4t 6 Are the x and acceleration components constant? Is acceleration a constant? Sample 14 For the rabbit in Example 1 & 13, find the acceleration a at time t = 15 s, in unit-vector notation and in magnitude-angle notation. 9

30 Sample 15 A particle with velocit v i ˆ.ˆ 4. j (in meters per second) at t = undergoes a constant acceleration a of magnitude a = 3. m/s at an angle θ = 13 from the positive direction of the x axis. What is the particle s velocit v at t = 5. s? PROJECTILE MOTION 3

31 Projectile Motion A projectile is a particle that moves in a vertical plane with some initial velocit v but its acceleration is alwas the free-fall acceleration g = 9.8 m/s. For the time being, we will ignore an effects due to air resistance. Checkpoint 9 Which of the following objects undergo projectile motion? (a) Tennis ball (b) Airplane (c) Diver (d) Frog (e) Bird (f) Rocket 31

32 Projectile Motion A projectile is launched with an initial velocit v v v xi ˆ v ˆj The components can then be found if we know the angle θ between v and the positive x direction. v v x v v cosq sinq Axes Independence During its two-dimensional motion, the projectile s position vector r and velocit vector v change continuousl, but its acceleration vector a is constant and alwas directed verticall downward. There is NO horizontal acceleration. In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other. Projectile motion can be broken into two separate onedimensional problems, one for the horizontal motion (constant motion problem) and one for the vertical motion (freefall/constant accelerated motion problem) 3

33 Checkpoint 1 At a certain instant, a fl ball has velocit v 5iˆ 4.9 ˆj (the x axis is horizontal, the axis is upward, and v is in meters per second). Has the ball passed its highest point? Projectile Motion Horizontal Because there is no acceleration in the horizontal direction, the horizontal component v x of the projectile s velocit remains unchanged from its initial value v x throughout the motion. x x x x v x t ( v cosq ) t 33

34 34 Projectile Motion Vertical The vertical motion is identical to freefall, and has a constant acceleration. ) ( ) sin ( sin 1 ) sin ( 1 g v v gt v v gt t v gt t v q q q Trajector Equation ) cos ( ) (tan q q v gx x

35 Horizontal Range The horizontal range R of a projectile is the horizontal distance the projectile has traveled when it returns to its initial (launch) height. v R g sin q The horizontal range R is maximum for a launch angle of 45, for these conditions. Caution: This equation does not give the horizontal distance traveled b a projectile when the final height is not the launch height Checkpoint 11 A fl ball is hit to the outfield. During its flight (ignore the effects of the air), what happens to its (a) horizontal and (b) vertical components of velocit? What are the (c) horizontal and (d) vertical components of its acceleration during ascent, during descent and at the topmost point of its flight? 35

36 Sample 16 A pirate ship is 56 m from a fort defending a harbor entrance. A defense cannon, located at sea level, fires balls at initial speed v = 8 m/s. (a) At what angle θ from the horizontal must a ball be fired to hit the ship? (b) What is the maximum range of the cannonballs? Sample 17 A rescue plane flies at 198 km/h and constant height h = 5 m toward a point directl over a victim, where a rescue capsule is to land. (a) What should be the angle φ of the pilot s line of sight to the victim when the capsule release is made? (b) As the capsule reaches the water, what is its velocit v in unit-vector notation and in magnitude-angle notation? 36

37 Sample 18 The figure illustrates the ramps for the current world-record motorccle jump, set b Jason Renie in. The ramps were H = 3. m high, angled at θ R = 1., and separated b distance D = 77. m. Assuming that he landed halfwa down the landing ramp and that the slowing effects of the air were negligible, calculate the speed at which he left the launch ramp. 37

38 UNIFORM CIRCULAR MOTION Uniform Circular Motion A particle is in uniform circular motion if it travels around a circle or a circular arc at constant (uniform) speed. Although the speed does not var, the particle is accelerating because the velocit changes in direction. This acceleration is alwas directed radiall inward, while the velocit vector at an instant is tangential to the circle. Due to this fact, the acceleration is known as centripetal center seeking acceleration. 38

39 Uniform Circular Motion Equations a c T v r r v Centripetal acceleration Period The period of revolution is the time for a particle to go around a closed path exactl once. 39

40 Checkpoint 1 An object moves at constant speed along a circular path in a horizontal x plane, with the center at its origin. When the object is at x = - m, its velocit is (4 m/s)ĵ. Give the object s (a) velocit and (b) acceleration at = m. Sample 19 Top gun pilots have long worried about taking a turn too tightl. As a pilot s bod undergoes centripetal acceleration, with head toward the center of the curvature, the blood pressure decreases, leading to loss of brain function. There are several warning signs to signal a pilot to ease up. When the centripetal acceleration is g or 3g, the pilot feels heav. At about 4g, the pilot s vision switches to back and white and narrows to tunnel vision. If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious a condition known as g-loc for g-induced loss of consciousness. What is the centripetal acceleration, in g units, of a pilot fling an F- at speed v = 5 km/h through a circular arc having a radius of curvature r = 5.8 km? 4

41 RELATIVE MOTION Relative Motion Suppose ou are riding on the school bus. You are watching the trees pass outside of our window. You know that ou are moving and the trees are stationar, but it looks like the trees are moving backward relative to ou. Science sas that EITHER of these interpretations, known as reference frames, are valid and the laws of phsics appl to them both. Most of the time, we choose a reference frame to be the ground because it is not moving. 41

42 Relative Motion in One Dimension Suppose that a police officer, Alex, is parked b the side of a highwa. We will place the origin of a stationar reference frame (Frame A) at this point. A second police officer, Barbara, is driving along the highwa at a constant speed. We add a second reference frame (Frame B). Frame B is moving with the same constant speed as Barbara so that she is alwas at the origin. Suddenl, a Porsche (P) speeds b and both officers measure the position of the car at the same instant. Relative Motion in One Dimension B interpreting the situation we can see that: The coordinate of the Porsche as measured b Alex is equal to the coordinate of the Porsche as measured b Barbara plus the coordinate of Barbara as measured b Alex. Or, if we take the derivative of the above equation: The velocit of the Porsche as measured b Alex is equal to the velocit of the Porches as measured b Barbara plus the velocit of Barbara as measured b Alex. v x PA PA x v If we take another derivative we find that both Alex and Barbara measure the same acceleration of the Porsche. a Observers on different frames of reference that move at constant velocit relative to each other will measure the same acceleration for a moving particle. PB PB x v PA a PB BA BA 4

43 Sample Suppose that Barbara s velocit relative to Alex is a constant v BA = 5 km/h and the Porsche (P) is moving in the negative direction of the x axis. (a) If Alex measures a constant v PA = -78 km/h for P, what velocit v PB will Barbara measure? (b) If P brakes to a stop relative to Alex in time t = 1 s at constant acceleration, what is its acceleration a PA relative to Alex? (c) What is the acceleration a PB of P relative to Barbara during the braking? Checkpoint 13 The table gives velocities (km/h) for Barbara and the Porsche for three situations. For each, what is the missing value and how is the distance between Barbara and the Porsche changing? Situation v BA v PA v PB (a) (b) (c) +6-43

44 Relative Motion in Two Dimensions rpa rpb rba vpa vpb vba a a We can analze relative motion in two dimensions in the same wa we did in one dimensions except that we must now use position and velocit vectors instead of just 1D coordinates. PA PB The same rule applies: Observers on different frames of reference that move at constant velocit relative to each other will measure the same acceleration for a moving particle. Sample 1 A plane moves due east while the pilot points the plane somewhat south of east, toward a stead wind that blows to the northeast. The plane has velocit v PW relative to the wind, with an airspeed (speed relative to the wind) of 15 km/h, directed at angle θ south of east. The wind has velocit v WG relative to the ground with speed 65. km/h, directed. east of north. What is the magnitude of the velocit v PG of the plane relative to the ground, what is θ? 44

45 Checkpoint 14 In Sample 19, suppose the pilot turns the plane to point it due east without changing the airspeed. Do the following magnitudes increase, decrease, or remain the same: (a) v PG,, (b) v PG,x, (c) v PG? (You can answer without computation.) 45