1-D and 2-D Motion Test Friday 9/8

Transcription

1 1-D and -D Motion Test Frida 9/8 3-1 Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocit, force, momentum A scalar has onl a magnitude. Some scalar quantities: mass, time, temperature 1

2 3- Addition of Vectors Graphical Methods For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates. 3- Addition of Vectors Graphical Methods Even if the vectors are not at right angles, the can be added graphicall b using the tail-to-tip method.

3 3-4 Adding Vectors b Components An vector can be epressed as the sum of two other vectors, which are called its components. Usuall the other vectors are chosen so that the are perpendicular to each other. 3-4 Adding Vectors b Components If the components are perpendicular, the can be found using trigonometric functions. 3

4 Sample Problem You run in a straight line at a velocit of 5.0 m/s in a direction that is 40 o south of west. a) What is the -component of the velocit? b) What is the -component of the velocit? -component 40 o -component 5.0m/s a)5.0m / s(cos 40 b)5.0m / s(sin 40 o o ) 3.8m / s ) 3.m / s Assignment Vector Activit on the class website Read pg. 4

5 1-D and -D Motion Test Frida 9/8 5

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8 You are adding vectors of length 0 and 40 units. What is the onl possible resultant magnitude that ou can obtain out of the following choices? A) 18 B) 37 C) 64 D) 100 A small cart is rolling at constant velocit on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? A) it depends on how fast the cart is moving B) it falls behind the cart C) it falls in front of the cart D) it falls right back into the cart E) it remains at rest 8

9 Now the cart is being pulled along a horizontal track b an eternal force (a weight hanging over the table edge) and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball? A) it falls behind the cart B) it falls in front of the cart C) it falls right back into the cart D) it remains at rest You drop a package from a plane fling at constant speed in a straight line. Without air resistance, the package will: A) quickl lag behind the plane while falling B) remain verticall under the plane while falling C) move ahead of the plane while falling D) not fall at all 9

10 -Dimensional Motion Definition: motion that occurs with both and components. Eample: Plaing pool. Throwing a ball to another person. Each dimension of the motion can obe different equations of motion. Solving -D Problems Resolve all vectors into components -component Y-component Work the problem as two one-dimensional problems. Each dimension can obe different equations of motion. Re-combine the results for the two components at the end of the problem. 10

11 Equations of Kinematics v v 1 o v at o v t v vo a vot 1 at v 1 v a t v v t o o v o t 1 at v vo a 11

12 v v o a t 1 v t o at t 1 v o v v vo a The part of the motion occurs eactl as it would if the part did not occur at all, and vice versa. 1

13 Eample In the direction, the spacecraft has an initial velocit component of + m/s and an acceleration of +4 m/s. In the direction, the analogous quantities are +14 m/s and an acceleration of +1 m/s. Find the final velocit of the spacecraft at time 7.0 s. v o a t 7.0s v v o 4 at 1 v 6.1m / s (6.8m / s v 14m / s m / s 6.8m / s )(7.0s) Sample Problem A roller coaster rolls down a 0 o incline with an acceleration of 5.0 m/s. a) How far horizontall has the coaster traveled in 10 seconds? b) How far verticall has the coaster traveled in 10 seconds? a a (5.0m / s )cos 0 4.7m / s (5.0m / s )sin 0 1.7m / s a) v 0 b) v 0 o 1 o t 1 t 1 1 a t a (4.7)(10 ) 35m t (1.7)(10 ) 35m 13

14 Projectiles Projectile Motion Something is fired, thrown, shot, or hurled near the earth s surface. Horizontal velocit is constant. Vertical velocit is accelerated. Air resistance is ignored. 14

15 1-Dimensional Projectile Definition: A projectile that moves in a vertical direction onl, subject to acceleration b gravit. Eamples: Drop something off a cliff. Throw something straight up and catch it. You calculate vertical motion onl. The motion has no horizontal component. -Dimensional Projectile Definition: A projectile that moves both horizontall and verticall, subject to acceleration b gravit in vertical direction. Eamples: Throw a softball to someone else. Fire a cannon horizontall off a cliff. Shoot a monke with a blowgun. You calculate vertical and horizontal motion. 15

16 Horizontal Component of Velocit Is constant Not accelerated Not influenced b gravit Follows equation: = V o t Horizontal Component of Velocit 16

17 Vertical Component of Velocit Undergoes accelerated motion Accelerated b gravit (9.8 m/s down) V = V o + gt = V o t + 1/gt V = V o + g Horizontal and Vertical 17

18 Solving -D Problems Resolve all vectors into components -component Y-component Work the problem as two one-dimensional problems. Each dimension can obe different equations of motion. Re-combine the results for the two components at the end of the problem. Equations of Kinematics v v 1 o v at o v t v vo a vot 1 at 18

19 The part of the motion occurs eactl as it would if the part did not occur at all, and vice versa. Eample In the direction, the spacecraft has an initial velocit component of + m/s and an acceleration of +4 m/s. In the direction, the analogous quantities are +14 m/s and an acceleration of +1 m/s. Find the final velocit of the spacecraft at time 7.0 s. v o a t 7.0s v v o 4 at 1 v 6.1m / s (6.8m / s v 14m / s m / s 6.8m / s )(7.0s) 19

20 Zero Launch Angle Projectiles Launch angle Definition: The angle at which a projectile is launched. The launch angle determines what the trajector of the projectile will be. Launch angles can range from -90 o (throwing something straight down) to +90 o (throwing something straight up) and everthing in between. 0

21 Zero Launch angle A zero launch angle implies a perfectl horizontal launch. v o Sample Problem The Zambezi River flows over Victoria Falls in Africa. The falls are approimatel 108 m high. If the river is flowing horizontall at 3.6 m/s just before going over the falls, what is the speed of the water when it hits the bottom? How far horizontall does the water travel? Assume the water is in freefall as it drops. v t v o 3.6m / s v speed g o (108m) 9.8m / s 4.7s gt 0 9.8(4.7) 46m / s m / s horizontal v t (3.6)(4.7) 16.9m 1

22 Assignment Pg Problems #1,,4,9,13,17,18,0 1-D and -D Motion Test Frida 9/8

23 Complete the following statement: In projectile motion, a) the horizontal motion depends on the vertical motion. b) the vertical motion depends on the horizontal motion. c) the horizontal acceleration depends on the vertical acceleration. d) the horizontal motion and the vertical motion are independent of each other. e) the vertical acceleration depends on the horizontal acceleration. A football is kicked at an angle 5 with respect to the horizontal. Which one of the following statements best describes the acceleration of the football during this event if air resistance is neglected? a) The acceleration is zero m/s at all times. b) The acceleration is zero m/s when the football has reached the highest point in its trajector. c) The acceleration is positive as the football rises, and it is negative as the football falls. d) The acceleration starts at 9.8 m/s and drops to some constant lower value as the ball approaches the ground. e) The acceleration is 9.8 m/s at all times. 3

24 Which one of the following statements concerning the range of a football is true if the football is kicked at an angle with an initial speed v 0? a) The range is independent of initial speed v 0. b) The range is onl dependent on the initial speed v 0. c) The range is independent of the angle. d) The range is onl dependent on the angle. e) The range is dependent on both the initial speed v 0 and the angle. Balls A, B, and C are identical. From the top of a tall building, ball A is launched with a velocit of 0 m/s at an angle of 45 above the horizontal direction, ball B is launched with a velocit of 0 m/s in the horizontal direction, and ball C is launched with a velocit of 0 m/s at an angle of 45 below the horizontal direction. Which of the following choices correctl relates the magnitudes of the velocities of the balls just before the hit the ground below? Ignore an effects of air resistance. a) v A = v C > v B b) v A = v C = v B c) v A > v C > v B d) v A < v C < v B e) v A > v C < v B 4

25 Zero Launch Angle Projectiles Launch angle Definition: The angle at which a projectile is launched. The launch angle determines what the trajector of the projectile will be. Launch angles can range from -90 o (throwing something straight down) to +90 o (throwing something straight up) and everthing in between. 5

26 Zero Launch angle A zero launch angle implies a perfectl horizontal launch. v o Sample Problem The Zambezi River flows over Victoria Falls in Africa. The falls are approimatel 108 m high. If the river is flowing horizontall at 3.6 m/s just before going over the falls, what is the speed of the water when it hits the bottom? How far horizontall does the water travel? Assume the water is in freefall as it drops. v t v o 3.6m / s v speed g o (108m) 9.8m / s 4.7s gt 0 9.8(4.7) 46m / s m / s horizontal v t (3.6)(4.7) 16.9m 6

27 You need to work on the projectile pre lab and must finish it b the start of class tomorrow. 1-D and -D Motion Test Frida 9/8 7

28 A baseball is hit upward and travels along a parabolic arc before it strikes the ground. Which one of the following statements is necessaril true? a) The velocit of the ball is a maimum when the ball is at the highest point in the arc. b) The -component of the velocit of the ball is the same throughout the ball's flight. c) The acceleration of the ball decreases as the ball moves upward. d) The velocit of the ball is zero m/s when the ball is at the highest point in the arc. e) The acceleration of the ball is zero m/s when the ball is at the highest point in the arc. 8

29 A ball is launched with an initial velocit v 0 as shown. Which one of the following arrows best represents the direction of the acceleration at point A? a) b) c) d) e) The acceleration at point A is zero m/s. General Launch Angle Projectiles 9

30 General launch angle v o Projectile motion is more complicated when the launch angle is not straight up or down (90 o or 90 o ), or perfectl horizontal (0 o ). General launch angle v o You must begin problems like this b resolving the velocit vector into its components. 30

31 Resolving the velocit Use speed and the launch angle to find horizontal and vertical velocit components V o V o = V o sin V o = V o cos Resolving the velocit Then proceed to work problems just like ou did with the zero launch angle problems. V o V o = V o sin V o = V o cos 31

32 Projectiles launched over level ground These projectiles have highl smmetric characteristics of motion. It is hand to know these characteristics, since a knowledge of the smmetr can help in working problems and predicting the motion. Lets take a look at projectiles launched over level ground. Trajector of a -D Projectile Definition: The trajector is the path traveled b an projectile. It is plotted on an - graph. 3

33 Trajector of a -D Projectile Mathematicall, the path is defined b a parabola. Trajector of a -D Projectile For a projectile launched over level ground, the smmetr is apparent. 33

34 Range of a -D Projectile Range Definition: The RANGE of the projectile is how far it travels horizontall. Maimum height of a projectile Maimum Height Range The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward. 34

35 Maimum height of a projectile Maimum Height Range The vertical velocit component is zero at maimum height. Maimum height of a projectile Maimum Height Range For a projectile launched over level ground, the maimum height occurs halfwa through the flight of the projectile. 35

36 Acceleration of a projectile g g g g g Acceleration points down at 9.8 m/s for the entire trajector of all projectiles. Velocit of a projectile v v v v o vf Velocit is tangent to the path for the entire trajector. 36

37 Velocit of a projectile v v v v v v v v v The velocit can be resolved into components all along its path. Velocit of a projectile v v v v v v v v v Notice how the vertical velocit changes while the horizontal velocit remains constant. 37

38 Velocit of a projectile v v v v v v v v v Maimum speed is attained at the beginning, and again at the end, of the trajector if the projectile is launched over level ground. Velocit of a projectile v o Launch angle is smmetric with landing angle for a projectile launched over level ground. v o - 38

39 Time of flight for a projectile t t o = 0 The projectile spends half its time traveling upward Time of flight for a projectile t t o = 0 t and the other half traveling down. 39

40 Sample problem A projectile fired from a gun has initial horizontal and vertical components of velocit equal to 30 m/s and 40 m/s, respectivel. Determine the initial speed of the projectile and determine the angle is the projectile fired (measured with respect to the horizontal). v m/s 40 tan o tan 53.1 Sample problem A soccer ball is kicked with a speed of 9.50 m/s at an angle of 5 o above the horizontal. If the ball lands at the same level from which is was kicked, how long was it in the air and how far did it travel? v v o 9.5sin m/s 4.01 m/s a 9.8 m/s v v at o ( 9.8) t t t 0.8 sec o v o 9.5cos m/s t 0.8 sec a 0 m/s v t (8.61)(0.8) 7.1 m o 40

41 Sample problem Snowballs are thrown with a speed of 13 m/s from a roof 7.0 m above the ground. Snowball A is thrown straight downward; snowball B is thrown in a direction 5 o above the horizontal. When the snowballs land, is the speed of A greater than, less than, or the same speed of B? The land with the same speed Position graphs for -D projectiles t t 41

42 Velocit graphs for -D projectiles V V t t Acceleration graphs for -D projectiles a a t t 4

43 The Range Equation Derivation is an important part of phsics. Your book has man more equations than our formula sheet. The Range Equation is in our tetbook, but not on our formula sheet. You can use it if ou can memorize it or derive it! The Range Equation R = v o sin( )/g. R: range of projectile fired over level ground v o : initial velocit g: acceleration due to gravit : launch angle 43

44 Assignment Do pg Questions #4,6,1 Problems #,4,6,31 44