Announcements. Unit 1 homework due tomorrow 11:59 PM Quiz 1 on 3:00P Unit 1. Units 2 & 3 homework sets due 11:59 PM

Transcription

1 Announcements Unit 1 homework due tomorrow 11:59 PM Quiz 1 on 3:00P Unit 1 Ø First 12 minutes of class: be on time!!! Units 2 & 3 homework sets due 11:59 PM Ø Most homework sets now due on Sunday Ø Check smartphysics Schedule!!! Mechanics Lecture 2, Slide 1

2 PhET Interactive Simulations Go to Supplementary Material page on Canvas Click on Motion under PhET Simulations Ø Vector Addition Ø Projectile Motion Ø Calculus Grapher Mechanics Lecture 2, Slide 2

3 Classical Mechanics Lecture 2: Vectors and 2-D Kinematics a) Vectors Today's Concepts: b) Projectile motion Mechanics Lecture 2, Slide 3

4 Unit 2 Learning Objectives If you master this unit, you should: be able to use the principle of superposition to separate the horizontal and vertical motion of a projectile in free fall. be able to identify the qualitative and quantitative relationship between a projectile s maximum height, time of flight, and initial velocity. be able to determine the horizontal and vertical position of a projectile at any point in time (equations of motion) given sufficient information about its trajectory (e.g., initial velocity, maximum height, ). be able to combine the horizontal and vertical equations of motion into a single vector equation (and vice versa). Mechanics Lecture 2, Slide 4

5 Projectile Motion In this class, a projectile is an object that moves only under the influence of uniform gravity (e.g., neglect air resistance; variations in gravity, etc.) Projectiles follow parabolic paths. The velocity of a projectile changes constantly during flight: it is ALWAYS accelerating. Mechanics Lecture 2, Slide 5

6 Projectile Motion & Frames of Reference We talked about both of these cases in Unit 1: state line Bonnie Cop 30 m/s Constant v Constant a d =1000 m Mechanics Lecture 2, Slide 6

7 Superposition v train car Since gravity acts downward only, motion along x and y are independent: Total motion is the superposition of motion along x and y solve each part separately! Mechanics Lecture 2, Slide 7

8 ACT A flatbed railroad car is moving along a track at constant velocity. A passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land? A) Forward of the center of the car B) At the center of the car C) Backward of the center of the car v train car Since a x = 0, the ball and the center of car have the same x position and x velocity throughout the motion!!! Mechanics Lecture 2, Slide 8

9 Equations of motion for a projectile gt 1 2 gt2 For projectile motion: Can arrange coordinate system to eliminate z Can align y-axis with gravity (vertical: up is + ) Ø a x = 0 Ø a y = -g = m/s 2 Mechanics Lecture 2, Slide 9

10 Example Problem (from Exam #1, Spring 2012) Lionel Messi kicks a soccer ball across a level field. The ball spends 3 seconds in the air and lands 60 m from the point where it was kicked. (Ignore air resistance.) What is the height of the ball at the peak of its trajectory? 1. Understand the Problem Read the problem carefully. Construct a mental image. Determine the question. Summarize the given information. 2. Describe the Physics Draw a useful diagram. Assign symbols to known and unknown quantities. Label relevant quantities on the diagram. Declare the target variable. State the relevant physical principles. Mechanics Lecture 2, Slide 10

11 Steps 1 & 2: Understand problem, describe physics Lionel Messi kicks a soccer ball across a level field. The ball spends 3 seconds in the air and lands 60 m from the point where it was kicked. (Ignore air resistance.) What is the height of the ball at the peak of its trajectory? t = 0 v 0 v 0y H t = 3 s v 0x 60m Mechanics Lecture 2, Slide 11

12 Step 2: (Physical Principles) Independent horizontal and vertical motion Ø Horizontal: constant velocity (a x = 0) Ø Vertical: constant acceleration (a y = -g) Horizontal distance traveled is determined by horizontal speed and time of flight Max height (at peak) is determined by initial vertical speed and acceleration due to gravity Time of flight usually due to vertical motion Mechanics Lecture 2, Slide 12

13 Vertical Motion 0 v y = v 0y + a y t = v 0y gt At peak height, what is v y? At peak: v 0y = gt =) t up = v 0 y g Max height: 0 y y 0 = v 0y t a yt 2 apple v0y =) H = v 0y g 1 2 g Time up/down depends only on initial vertical velocity! 1 =) H 0=v 0y t up 2 gt2 up apple v0y g 2 = v2 0 y 2g Max height also depends only on initial vertical velocity! Mechanics Lecture 2, Slide 13

14 Checkpoint: Three Ships (part 2) A destroyer fires two shells with different ini5al speeds at two different enemy ships. The shells follow the trajectories shown. Which enemy ship gets hit first? Destroyer Enemy 1 Enemy 2 A) Enemy 1 B) Enemy 2 C) They are both hit at the same time Time up/down will be the same for objects traveling the same ver5cal distance!!! Mechanics Lecture 2, Slide 14

15 Checkpoint: Three Ships (part 1) A destroyer simultaneously fires two shells with the same ini5al speed at two different enemy ships. The shells follow the trajectories shown. Which ship gets hit first. Destroyer Enemy 1 Enemy 2 A) Enemy 1 B) Enemy 2 C) They are both hit at the same 5me Time up/down and height depend only initial vertical speed: Higher è longer time of flight Mechanics Lecture 2, Slide 15

16 General Problem Solving Framework 1. Understand the Problem Read the problem carefully. Construct a mental image. Determine the question. Summarize the given information. 2. Describe the Physics Draw a useful diagram. Assign symbols to known and unknown quantities. Label relevant quantities on the diagram. Declare the target variable. State the relevant physical principles. 3. Plan the Solution Write down relevant equations. Write down relevant constraints. State the necessary approximations. Outline how to use the equations to determine the target variable. 4. Execute the Plan Solve for the target variable symbolically. Check the units of the equation. Substitute numerical values of known quantities. Calculate the value of the target variable. Answer the question. 5. Evaluate the Solution Is the answer properly stated? Is the answer reasonable? Is the answer complete? Mechanics Lecture 2, Slide 16

17 What is H? Step 3: Plan Solution t = 0 v 0 v 0y H t = 3 s v 0x Equations: v y = v 0y + a y t =) t up = v 0 y g Constraints: y y 0 = v 0y t a yt 2 =) H = v2 0 y 2g Time of flight = 3 s =) t up =1.5 s Approximations: Neglect drag (air resistance): gravity only force Outline: Find H in terms of t up and then plug in values (with units) Mechanics Lecture 2, Slide 17

18 General Problem Solving Framework 1. Understand the Problem Read the problem carefully. Construct a mental image. Determine the question. Summarize the given information. 2. Describe the Physics Draw a useful diagram. Assign symbols to known and unknown quantities. Label relevant quantities on the diagram. Declare the target variable. State the relevant physical principles. 3. Plan the Solution Write down relevant equations. Write down relevant constraints. State the necessary approximations. Outline how to use the equations to determine the target variable. 4. Execute the Plan Solve for the target variable symbolically. Check the units of the equation. Substitute numerical values of known quantities. Calculate the value of the target variable. Answer the question. 5. Evaluate the Solution Is the answer properly stated? Is the answer reasonable? Is the answer complete? Mechanics Lecture 2, Slide 18

19 Step 4: Execute the Plan t = 0 v 0 v 0x v 0y H t = 3 s Solve for target variable: Check units: Substitute numerical value: Calculate value and answer question: What is the value of H? A. 5 meters B. 7 meters C. 9 meters D. 11 meters E. 13 meters Mechanics Lecture 2, Slide 19

20 What is H? Step 4: Execute the Plan t = 0 v 0 v 0y H t = 3 s v 0x Solve for target variable: t up = v 0 y g =) v 0y = gt up Check units: [H] Substitute numerical value: H = v2 0 y 2g h m/s 2i s 2 [m] H =0.5 = g2 t 2 up 2g = gt2 up m/s 2 (1.5 s) 2 Calculate value and answer question: H = 11 m Mechanics Lecture 2, Slide 20

21 What is H? t = 0 Step 5: Evaluate the Solution v 0 v 0y v 0x H = 11 m t = 3 s Is the answer properly stated? Is the answer reasonable? Is the answer complete? Mechanics Lecture 2, Slide 21

22 y Equations of Motion t = 0 v 0 v Horizontal: v 0y H = 11 m t = 3 s 60m v x (t) =v 0x + a x t = v 0x x(t) =x 0 + v 0x t a xt 2 = v 0x t x Vertical: v y (t) =v 0y + a y t = v 0y y(t) =y 0 + v 0y t a yt 2 1 = v 0y t 2 gt2 To complete the equations of motion, we need to find initial velocity components. -g 0 -g gt Mechanics Lecture 2, Slide 22

23 ACT! Lionel Messi kicks a soccer ball across a level field. The ball spends 3 seconds in the air and lands 60 m from the point where it was kicked. (Ignore air resistance.) t = 0 v 0 v 0y H = 11 m t = 3 s v 0x 60m What is the horizontal speed of the ball just after being kicked? A. v 0x = 20.0 m/s B. v 0x = 14.7 m/s C. v 0x = 24.8 m/s D. v 0x = 34.7 m/s x(t) =v 0x t =) x(t =3s)=v 0x (3 s) = 60 m =) v 0x = 60 m = 20 m/s 3s Mechanics Lecture 2, Slide 23

24 ACT! Lionel Messi kicks a soccer ball across a level field. The ball spends 3 seconds in the air and lands 60 m from the point where it was kicked. (Ignore air resistance.) t = 0 v 0 v 0y H = 11 m t = 3 s v 0x 60m What is the vertical speed of the ball just after being kicked? A. v 0y = 20.0 m/s B. v 0y = 14.7 m/s C. v 0y = 24.8 m/s D. v 0y = 34.7 m/s H = v2 0 y 2g =) v 0 y = p 2gH q = 2(9.8 m/s 2 )(11 m) = 14.7 m/s Mechanics Lecture 2, Slide 24

25 y Equations of Motion t = 0 v 0 v v 0y H = 11 m 60m t = 3 s x Horizontal: v x (t) = 20 m/s x(t) = (20 m/s) t Vertical: v y (t) = (14.7 m/s) y(t) = (14.7 m/s) t gt 1 2 gt2 Mechanics Lecture 2, Slide 25

26 Projectile motion and vectors During its flight, a projectile s velocity vector will change continuously: it undergoes constant (uniform) acceleration At any point in time, the velocity vector will point in its direction of motion The velocity vector can be broken into two independent components: horizontal and vertical We need to review a little about vectors Mechanics Lecture 2, Slide 26

27 Vectors A y A! A x Think of a vector as an arrow. (An object having both magnitude and direc5on) Mechanics Lecture 2, Slide 27

28 Vectors θ A = A A A x y = Acosθ = Asinθ tanθ = A y / A x Think of a vector as an arrow. (An object having both magnitude and direc5on) The object is the same no ma:er how we chose to describe it Mechanics Lecture 2, Slide 28

29 Computing the magnitude of a vector ~V To find the magnitude of a vector, use the quadratic formula: V y V x V = ~ V = q Vx 2 + Vy 2 + Vz 2 The magnitude of the velocity vector is called speed. (We don t have a special name for the magnitude of acceleration or position/displacement) Mechanics Lecture 2, Slide 29

30 ACT! Lionel Messi kicks a soccer ball across a level field. The ball spends 3 seconds in the air and lands 60 m from the point where it was kicked. (Ignore air resistance.) t = 0 v 0 v 0y H = 11 m t = 3 s v 0x 60m What is the total speed of the ball just after being kicked? A. v 0 = 20.0 m/s B. v 0 = 14.7 m/s C. v 0 = 24.8 m/s D. v 0 = 34.7 m/s v 0 = q v 2 0 x + v 2 0 y = 24.8 m/s Mechanics Lecture 2, Slide 30

31 ACT! Lionel Messi kicks a soccer ball across a level field. The ball spends 3 seconds in the air and lands 60 m from the point where it was kicked. (Ignore air resistance.) t = 0 v 0 v 0x v 0y H t = 3 s 60m What is the initial launch angle of the ball with respect to the ground? A B C = tan 1 v 0y /v 0x = 36.3 D Mechanics Lecture 2, Slide 31

32 Vector Addition Mechanics Lecture 2, Slide 32

33 General Properties of Vectors ~A = ha x,a y,a z i ~ B = hbx,b y,b z i Vector addition: ~C = ~ A + ~ B hc x,c y,c z i = ha x + B x,a y + B y,a z + B z i Product of a scalar and vector: M ~ A = M ha x,a y,a z i = hma x,ma y,ma z i Dot product (scalar product) of two vectors: ~A ~B = A x B x + A y B y + A z B z Mechanics Lecture 2, Slide 33

34 A! ACT: Vector Addition B! A! B! A! B! Vectors and are shown to the right. Which of the following best describes +? Add vectors 5p to tail: A B C D E Mechanics Lecture 2, Slide 34

35 A! ACT: Vector Subtraction B! A! B! A! B! Vectors and are shown to the right. Which of the following best describes -? Add vectors 5p to tail: A B C D E Mechanics Lecture 2, Slide 35

36 A! ACT: Vector Addition, part II B! A! 2B! A! B! Vectors and are shown to the right. Which of the following best describes +? Add vectors 5p to tail: A B C D E Mechanics Lecture 2, Slide 36

37 ACT: Monkey Troubles You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. In order to hit the monkey with the dart, where should you point the gun before shoo5ng? A) Right at the monkey B) Below the monkey C) Above the monkey Mechanics Lecture 2, Slide 37

38 Hitting the Ball Bullet x = v o t 1 y = gt 2 2 Ball x = x o 1 y = gt 2 2 Mechanics Lecture 2, Slide 38

39 Shooting the Cougar S5ll works even if you shoot upwards! Cougar x = x 0 1 y = y 0 2 gt2 Bullet x = v 0x t y = v 0y t 1 2 gt2 Bullet hits Cougar!! Mechanics Lecture 2, Slide 39

40 ACT: Monkey Troubles You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. In order to hit the monkey with the dart, where should you point the gun before shoo5ng? A) Right at the monkey B) Below the monkey C) Above the monkey Mechanics Lecture 2, Slide 40

41 General Problem Solving Framework 1. Understand the Problem Read the problem carefully. Construct a mental image. Determine the question. Summarize the given information. 2. Describe the Physics Draw a useful diagram. Assign symbols to known and unknown quantities. Label relevant quantities on the diagram. Declare the target variable. State the relevant physical principles. 3. Plan the Solution Write down relevant equations. Write down relevant constraints. State the necessary approximations. Outline how to use the equations to determine the target variable. 4. Execute the Plan Solve for the target variable symbolically. Check the units of the equation. Substitute numerical values of known quantities. Calculate the value of the target variable. Answer the question. 5. Evaluate the Solution Is the answer properly stated? Is the answer reasonable? Is the answer complete? Mechanics Lecture 2, Slide 41