Arial Bombing Techniques
|
|
- Miles Morton
- 5 years ago
- Views:
Transcription
1 Arial Bombing Techniques By Crystal Pepper and Chris Wilson March 30, 2009 Abstract In this article we will explore the bombing techniques used by the United States Army during World War II in order to produce a maximum impact on targets. We will demonstrate through calculations of differential equations how the speed of the plane, linear distance the plane is from the target, and altitude of the plane determines the time that the bombs must be deployed. Intro During World War II the role of the bombardier was critical in the accuracy of placing the free falling bombs on target. The conditions needed were: altitude, true airspeed, bomb ballistics, trail, actual time of fall, ground speed, and drift. The altitude, which is controlled by the pilot, determines the amount of time the bomb is free falling. The pilot also controls the true airspeed which gives the bomb its initial forward velocity and in turn affects the trail (horizontal distance that the bomb is behind the plane upon impact). The bomb ballistics such as the size, shape, and density determine the air resistance the bomb will encounter as it falls. The actual time of fall is the length of time the bomb is sustained in the air from the instant of release to the instant of impact. The actual time of fall is affected by the altitude and bomb ballistics. The ground speed is the speed of the plane in relation to the Earth s surface. This affects the range of the bomb. Lastly, the drift is determined by the direction and velocity of the wind and affects the distance the bomb will travel downwind. With the help of a mechanical analog computer, the then top secret Norden bomb sight, the bombardier was able to input the parameters altitude, airspeed, bomb ballistics, and air density. With these parameters, the accuracy of the placement of the bombs was sufficient for the technology available at the time. 1 Projectile Motion When the bomb leaves the airplane it will not fall in a straight line. From the moment the bomb leaves the plane its trajectory will vary in curvature. Projectile motion is two-dimensional motion. However, any case of two dimensional motion can be resolved into two cases of one dimensional motion: one along the x axis and the other along the y-axis. We will initially study the one dimensional cases separately, without considering air resistance. 1
2 Figure 1: Vertical Motion 1.1 Vertical Motion We now develop a mathematical model of the bomb s trajectory. For convenience, we will thin of our bomb as an object moving in the xy plane and mae use of this idea in our graphical analyses. Figure 1 shows the force on an object with a vertical motion. It is always true for an object near the Earth s surface that the acceleration of gravity is equal to the differential change in velocity in a given time interval. dv y dt = a = g We now that gravity g is 9.8, m/s 2. In our coordinate system, with y upward, this acceleration is a = g y = 9.8 m/s 2 (1) This is the acceleration along the y-axis. The velocity in the y direction can now be determined by taing the anti-derivative of g with respect to time. In order to set up our velocity equation, two initial conditions will be required. Our parameters are the initial time t = 0, and the initial vertical velocity v y = 0. dy dt = gdt v yf = gt + c, where c is a constant of integration. By letting c = v yi and v yi = 0 then it follows that v yf = v yi g y t v yf = 0 gt v yf = gt Equation (2) can be used to solve for the velocity along the y-axis at any instant t. By the same process, we can determine the displacement along the y-axis at any instant t. Integrating the velocity with respect to time gives (2) 2
3 y = (v yi gt)dt y = v yi t 1 2 g yt 2 y = g yt 2 (3) y = 1 2 gt2. We have now described all of the components of the bomb s vertical drop. We can now calculate the acceleration at any time, the velocity at any time, and the position at any time. 1.2 Horizontal Motion In the previous section, we described the motion of the bomb from its acceleration to its final displacement by taing the anti-derivative of each equation. Determining the velocity and displacement of the bomb in the horizontal direction can be calculated in the same manner. First, we will need to establish an initial condition. Since the bomb is attached to a plane which is flying in a horizontal direction, it will have the same initial velocity as the plane. However, due to air resistance and gravitation, the bomb will fall in an arc lie path towards the ground. In other words, there is no horizontal acceleration and the plane maintains its velocity from the time of release to impact. Therefore, the initial conditions are a x = 0 and v xi = v x. We will need to use a x to solve for our final displacement. Now the component of initial velocity along the x-axis can be written by taing the integral of the acceleration component dx dt = a x dt v xf = a x t + c (4) v xf = 0 + c v xf = c. The c represents an arbitrary constant which we can solve for by plugging in our initial values of v xi = v xf and a x = 0 v xi = a x t + c v xi = 0 + c v xi = c. Plugging this c value into equation (4) with the initial conditions, will give us a first order differential equation that describes the horizontal speed of our bomb. v xf = a x t + c v xf = a x t + v xi v xf = 0 + v xi v xf = v xi. 3
4 Figure 2: Horizontal Motion This is the component of the velocity along the x-axis at any instant t. This means that the horizontal component of velocity does not change throughout the projectile s motion. Let v xf = v xi = v x (5) represent our final velocity in the horizontal direction. By incorporating a horizontal acceleration, and integrating once more, we will be able to determine the displacement of the bombs trajectory. x f = (a x t + v x )dt x f = 1 2 a xt 2 + v x t + c. We will need to solve for c to determine the final displacement equation. Plugging in the initial values and letting t = 0 we will solve for the initial horizontal position x i = 1 2 a xt 2 + v x t + c x i = c x i = c. With this c value the displacement along the x-axis at any instant t can be described by x f = a x t 2 + v x + c x f = 0 + v x t + x i x f = v x t + x i. We can now determine the velocity and displacement at any time during the flight of the bomb. With equation (6), we can plot the bomb s trajectory on an xy coordinate system as shown in Figure 3. 2 The Real Forces of Projectile Motion 2.1 Drag and the Vertical Motion In sections 1.1 and 1.2 we showed you the idealized formulas for the motion of a projectile by neglecting the force of drag. However, in order to establish a more real world basis for our model, we will need to determine and incorporate this force. Drag is defined as the force that opposes the motion of an object in a fluid. This fluid can be anything from liquid to gas. As our bomb leaves (6) 4
5 Figure 3: Horizontal and Vertical trajectory. the plane, the air that it moves through will provide this resistance. Drag R depends on the density of the air ρ, the velocity v, the air s viscosity and compressibility D, and the reference area A. R = 1 2 DρAv2. (7) To derive the drag equation is a tedious tas that would tae hours of computation. So for the sae of time we state that this equation is just the drag force without any further explanation. Since air resistance is present in all real applications, the motion of the vertical fall of the bomb will now have a drag force. Using Newtons law of force F = ma we will sum the forces in the y direction to derive our acceleration formula. F = F g F d. Knowing that F = ma, F g = mg, and F d = 1 2DρAv, the equation loos lie ma = mg 1 2 DρAv a = g DρAv 2m. (8) Equation (8)includes all of the forces that are acting on a bomb as it falls to the ground. Before we attempt to integrate this equation, we will simplify the drag by letting = DρA/2m. This will help eep our values straight. Therefore, with the new substitution, equation (8) becomes 5
6 a = g v a = dv dt = g v. (9) Equation (9) is the acceleration due to the velocity and drag. By integrating this function, we will get a formula for the velocity with respect to time. In order for us to differentiate equation (9), we will need to separate the variables as follows dv dt = g v dv = (g v) dt dv g v = dt Now that the function is set up, we will integrate dv g v = dt ln (g v) = t + C ln (g v) = (t + C) e ln(g v) = e t+c g v = e t e C Through the integration, we are left with an arbitrary constant e c. To simplify this, let b = e c. With the substitution we can now solve for the velocity as a function of time. g v = be t v = g be t v = g b e t. However, this equation is not yet correct. We will need to substitute our initial values to solve for the constant b. Setting v = 0 at t = 0 produces v = g b e t 0 = g b e (0) 0 = g b g = b. Our calculations reveal that b is equal to g. This will mae the velocity equation a little bit easier to loo at. 6
7 v = g g e t v = g ( t ) (10) v = g ( t ). This formula tells us that at any time t we can evaluate the velocity of the bomb. We would lie to note that as t, v g/, called the terminal velocity. Up to this point, we have formulated equations for the acceleration and velocity of the bomb in the vertical direction. Neither of these equations, however, will tell us where the bomb will land when a given height and time are nown. We will integrate once more to find the displacement of the bomb v = g ( t ) dy dt = g ( t ) dy = g (1 e t ) dt y = g ) (t + e t + c. Yet again, we are left with a constant of integration c. Providing some initial conditions will eliminate the constant. With the initial condition y = y 0 at t = 0 gives y 0 = g ) (0 + e0 + c y 0 = g ( ) 1 + c y 0 = g 2 + c c = y 0 g 2. Plugging in the c value will give the final equation for displacement y = g ) (t + e t + y 0 g 2 y = g t + g 2 e t + y 0 g 2 (11) y = y 0 + g t g 2 ( t ). This final equation represents the general solution to our system in the vertical direction. We can plot this, as shown in figure 4, given any initial height of the plane. We have now established formulas for the acceleration, velocity, and position for our trajectory in the vertical direction. We will then follow up with the derivation for the equations with drag in the horizontal direction. 7
8 Figure 4: Vertical Motion with Drag 2.2 Drag and the Horizontal Motion In section 2.1, we derived the the general solution to the vertical displacement by integration. In the same way, we can derive the general solution to the horizontal dislpacement. By taing the same steps of integration we can go through this process without too much explanation. Starting with the initial forces in the x direction F = F d. (12) We can see from equation (12) that the only force in the horizontal direction is the force of drag. Solving this equation will provide a formula for the velocity. For convienance, we will let = DρA/2m for our drag force. a = dv dt = v Seperate variables then integrate to solve for the final velocity equation 8
9 dv dt = v dv v = dt dv v = dt dv v = dt lnv = t + c v = e t+c v = c 1 e t. The integration constant c 1 can be evaluated with the initial conditions t = 0 and v = v 0. (13) v = c 1 e t v 0 = c 1 e 0 v 0 = c 1. Then plugging this value bac into the equation (13) will give v = v 0 e t (14) This is the final eqaution for the velocity with drag. We will then integrate, just as we did with the vertical equation, to solve for the displacement. v = v 0 e t dx dt = v 0e t dx = v 0 e t dt x = v 0 e t dt x = v 0 e t + c. Yet again we are left with the integral constant that can be solved with the initial conditions t = 0 and x = 0 x = v 0 e t + c 0 = v 0 e 0 + c 0 = v 0 + c v 0 = c 9
10 Figure 5: Horizontal Motion with Drag Therefore e t x = v 0 + v 0 x = v 0 ( t ) (15) Equation (15) is our final formula needed to solve for the discplacement of the bomb on the x-axis. Setting an altitude and initial speed of the plane, we can plot this trajectory as shown in figure 5. 3 Conclusion With equations (11) and (15), y = y 0 + g t g 2 ( t ) x = v 0 ( t ) we can plot the trajectory of the bomb in an xy coordinate system. These equations can be used to solve the position of the bomb when dropped from any height and distance. Setting an altitude and velocity of the plane, we have plotted these equations as demonstrated in figure 6. 10
11 Figure 6: Horizontal and Vertical Motion with Drag References [1] Probable Bomb Impact Areas (PJR) I04FL [2] Research and Education Association Differential Equations. 11
Motion Along a Straight Line (Motion in One-Dimension)
Chapter 2 Motion Along a Straight Line (Motion in One-Dimension) Learn the concepts of displacement, velocity, and acceleration in one-dimension. Describe motions at constant acceleration. Be able to graph
More informationSolutions to Homework 1, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y(x) = ce 2x + e x
Solutions to Homewor 1, Introduction to Differential Equations, 3450:335-003, Dr. Montero, Spring 2009 problem 2. The problem says that the function yx = ce 2x + e x solves the ODE y + 2y = e x, and ass
More information3.4 Projectile Motion
3.4 Projectile Motion Projectile Motion A projectile is anything launched, shot or thrown---i.e. not self-propelled. Examples: a golf ball as it flies through the air, a kicked soccer ball, a thrown football,
More informationMotion in Two Dimensions. 1.The Position, Velocity, and Acceleration Vectors 2.Two-Dimensional Motion with Constant Acceleration 3.
Motion in Two Dimensions 1.The Position, Velocity, and Acceleration Vectors 2.Two-Dimensional Motion with Constant Acceleration 3.Projectile Motion The position of an object is described by its position
More informationProblem: Projectile (CM-1998)
Physics C -D Kinematics Name: ANSWER KEY AP Review Packet Vectors have both magnitude and direction displacement, velocity, acceleration Scalars have magnitude only distance, speed, time, mass Unit vectors
More informationWhen we throw a ball :
PROJECTILE MOTION When we throw a ball : There is a constant velocity horizontal motion And there is an accelerated vertical motion These components act independently of each other PROJECTILE MOTION A
More information2. KINEMATICS. By Liew Sau Poh
2. KINEMATICS By Liew Sau Poh 1 OBJECTIVES 2.1 Linear motion 2.2 Projectiles 2.3 Free falls and air resistance 2 OUTCOMES Derive and use equations of motion with constant acceleration Sketch and use the
More informationChapter 4. Motion in Two Dimensions. Position and Displacement. General Motion Ideas. Motion in Two Dimensions
Motion in Two Dimensions Chapter 4 Motion in Two Dimensions Using + or signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motion
More informationAH Mechanics Checklist (Unit 1) AH Mechanics Checklist (Unit 1) Rectilinear Motion
Rectilinear Motion No. kill Done 1 Know that rectilinear motion means motion in 1D (i.e. along a straight line) Know that a body is a physical object 3 Know that a particle is an idealised body that has
More informationProblem: Projectile (CM-1998) Justify your answer: Problem: Projectile (CM-1998) 5 10 m/s 3. Show your work: 3 m/s 2
Physics C -D Kinematics Name: AP Review Packet Vectors have both magnitude and direction displacement, velocity, acceleration Scalars have magnitude only distance, speed, time, mass Unit vectors Specify
More information34.3. Resisted Motion. Introduction. Prerequisites. Learning Outcomes
Resisted Motion 34.3 Introduction This Section returns to the simple models of projectiles considered in Section 34.1. It explores the magnitude of air resistance effects and the effects of including simple
More informationBell Ringer: What is constant acceleration? What is projectile motion?
Bell Ringer: What is constant acceleration? What is projectile motion? Can we analyze the motion of an object on the y-axis independently of the object s motion on the x-axis? NOTES 3.2: 2D Motion: Projectile
More informationMOTION IN TWO OR THREE DIMENSIONS
MOTION IN TWO OR THREE DIMENSIONS 3 Sections Covered 3.1 : Position & velocity vectors 3.2 : The acceleration vector 3.3 : Projectile motion 3.4 : Motion in a circle 3.5 : Relative velocity 3.1 Position
More informationPHY 1114: Physics I. Quick Question 1. Quick Question 2. Quick Question 3. Quick Question 4. Lecture 5: Motion in 2D
PHY 1114: Physics I Lecture 5: Motion in D Fall 01 Kenny L. Tapp Quick Question 1 A child throws a ball vertically upward at the school playground. Which one of the following quantities is (are) equal
More informationChapter 3 Acceleration
Chapter 3 Acceleration Slide 3-1 Chapter 3: Acceleration Chapter Goal: To extend the description of motion in one dimension to include changes in velocity. This type of motion is called acceleration. Slide
More informationIn this activity, we explore the application of differential equations to the real world as applied to projectile motion.
Applications of Calculus: Projectile Motion ID: XXXX Name Class In this activity, we explore the application of differential equations to the real world as applied to projectile motion. Open the file CalcActXX_Projectile_Motion_EN.tns
More informationKINEMATICS OF A PARTICLE. Prepared by Engr. John Paul Timola
KINEMATICS OF A PARTICLE Prepared by Engr. John Paul Timola Particle has a mass but negligible size and shape. bodies of finite size, such as rockets, projectiles, or vehicles. objects can be considered
More informationChapter 3 Motion in two or three dimensions
Chapter 3 Motion in two or three dimensions Lecture by Dr. Hebin Li Announcements As requested by the Disability Resource Center: In this class there is a student who is a client of Disability Resource
More informationKINEMATICS. Challenging MCQ questions by The Physics Cafe. Compiled and selected by The Physics Cafe
KINEMATICS Challenging MCQ questions by The Physics Cafe Compiled and selected by The Physics Cafe 1 Two diamonds begin free fall from rest from the same height 1.0 s apart. How long after the first diamond
More informationProjectile Motion I. Projectile motion is an example of. Motion in the x direction is of motion in the y direction
What is a projectile? Projectile Motion I A projectile is an object upon which the only force acting is gravity. There are a variety of examples of projectiles. An object dropped from rest is a projectile
More informationAxis Balanced Forces Centripetal force. Change in velocity Circular Motion Circular orbit Collision. Conservation of Energy
When something changes its velocity The rate of change of velocity of a moving object. Can result from a change in speed and/or a change in direction On surface of earth, value is 9.8 ms-²; increases nearer
More informationVectors and Projectile Motion on the TI-89
1 Vectors and Projectile Motion on the TI-89 David K. Pierce Tabor Academy Marion, Massachusetts 2738 dpierce@taboracademy.org (58) 748-2 ext. 2243 This paper will investigate various properties of projectile
More information8.6 Drag Forces in Fluids
86 Drag Forces in Fluids When a solid object moves through a fluid it will experience a resistive force, called the drag force, opposing its motion The fluid may be a liquid or a gas This force is a very
More informationChapter 4. Motion in Two Dimensions
Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail Will treat projectile motion and uniform circular motion
More informationFree Falling Objects and Reynolds Numbers
Free Falling Objects and Nate Currier May 12, 2009 Page 1 of 19 Abstract This paper will discuss the underlying physics of the forces acting on a free-falling object, and the drag forces acting on those
More informationHonors Physics Acceleration and Projectile Review Guide
Honors Physics Acceleration and Projectile Review Guide Major Concepts 1 D Motion on the horizontal 1 D motion on the vertical Relationship between velocity and acceleration Difference between constant
More informationChapter 2. Motion in One Dimension
Chapter 2 Motion in One Dimension Types of Motion Translational An example is a car traveling on a highway. Rotational An example is the Earth s spin on its axis. Vibrational An example is the back-and-forth
More information5 Projectile Motion. Projectile motion can be described by the horizontal and vertical components of motion.
Projectile motion can be described by the horizontal and vertical components of motion. In the previous chapter we studied simple straight-line motion linear motion. Now we extend these ideas to nonlinear
More informationLesson 2. Physics 168. Luis Anchordoqui
Lesson 2 Physics 168 Luis Anchordoqui Deriving Constant-Acceleration Kinematic Equations To obtain an equation for position as a function of time! look at special case of motion with constant velocity!
More informationy(t) = y 0 t! 1 2 gt 2. With y(t final ) = 0, we can solve this for v 0 : v 0 A ĵ. With A! ĵ =!2 and A! = (2) 2 + (!
1. The angle between the vector! A = 3î! 2 ĵ! 5 ˆk and the positive y axis, in degrees, is closest to: A) 19 B) 71 C) 90 D) 109 E) 161 The dot product between the vector! A = 3î! 2 ĵ! 5 ˆk and the unit
More informationForces and Motion in One Dimension
Nicholas J. Giordano www.cengage.com/physics/giordano Forces and Motion in One Dimension Applications of Newton s Laws We will learn how Newton s Laws apply in various situations We will begin with motion
More informationLecture XXVI. Morris Swartz Dept. of Physics and Astronomy Johns Hopkins University November 5, 2003
Lecture XXVI Morris Swartz Dept. of Physics and Astronomy Johns Hopins University morris@jhu.edu November 5, 2003 Lecture XXVI: Oscillations Oscillations are periodic motions. There are many examples of
More informationGeneral Physics I. Lecture 3: Newton's Laws. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 3: Newton's Laws Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ What Causes Changes of Motion? We define the interaction of a body with its environment
More informationChapter 10. Projectile and Satellite Motion
Chapter 10 Projectile and Satellite Motion Which of these expresses a vector quantity? a. 10 kg b. 10 kg to the north c. 10 m/s d. 10 m/s to the north Which of these expresses a vector quantity? a. 10
More informationAnnouncement. Quiz on Friday (Graphing and Projectile Motion) No HW due Wednesday
Going over HW3.05 Announcement Quiz on Friday (Graphing and Projectile Motion) No HW due Wednesday As the red ball rolls off the edge, a green ball is dropped from rest from the same height at the same
More information3.2 Projectile Motion
Motion in 2-D: Last class we were analyzing the distance in two-dimensional motion and revisited the concept of vectors, and unit-vector notation. We had our receiver run up the field then slant Northwest.
More informationClass 11 Physics NCERT Exemplar Solutions Motion in a Straight Line
Class 11 Physics NCERT Exemplar Solutions Motion in a Straight Line Multiple Choice Questions Single Correct Answer Type Q1. Among the four graphs shown in the figure, there is only one graph for which
More informationChapter 3 Acceleration
Chapter 3 Acceleration Slide 3-1 PackBack The first answer gives a good physical picture. The video was nice, and worth the second answer. https://www.youtube.com/w atch?v=m57cimnj7fc Slide 3-2 Slide 3-3
More informationBreak problems down into 1-d components
Motion in 2-d Up until now, we have only been dealing with motion in one-dimension. However, now we have the tools in place to deal with motion in multiple dimensions. We have seen how vectors can be broken
More informationNotes 4: Differential Form of the Conservation Equations
Low Speed Aerodynamics Notes 4: Differential Form of the Conservation Equations Deriving Conservation Equations From the Laws of Physics Physical Laws Fluids, being matter, must obey the laws of Physics.
More informationChapter 4. Motion in Two Dimensions
Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail Will treat projectile motion and uniform circular motion
More informationPhysics 8 Monday, September 9, 2013
Physics 8 Monday, September 9, 2013 HW2 (due Friday) printed copies. Read Chapter 4 (momentum) for Wednesday. I m reading through the book with you. It s my 3rd time now. One purpose of the reading responses
More informationPS 11 GeneralPhysics I for the Life Sciences
PS 11 GeneralPhysics I for the Life Sciences M E C H A N I C S I D R. B E N J A M I N C H A N A S S O C I A T E P R O F E S S O R P H Y S I C S D E P A R T M E N T N O V E M B E R 0 1 3 Definition Mechanics
More information2-D Vector Equations have the same form as 1-D Kinematics. f i i
2-D Vector Equations have the same form as 1-D Kinematics v = v + at f i 1 r = r + v t+ at f i i 2 2 2-D Vector Equations have the same form as 1-D Kinematics v = viˆ+ v ˆj f x y = ( v + ati ) ˆ+ ( v +
More informationChapter 3. Motion in One Dimension
Chapter 3 Motion in One Dimension Outline 3.1 Position, Velocity and Speed 3.2 Instantaneous Velocity and Speed 3.3 Acceleration 3.4 Motion Diagrams 3.5 One-Dimensional Motion with Constant Acceleration
More informationPHYS 211 Lecture 9 - Examples of 3D motion 9-1
PHYS 211 Lecture 9 - Examples of 3D motion 9-1 Lecture 9 - Examples of 3D motion Text: Fowles and Cassiday, Chap. 4 In one dimension, the equations of motion to be solved are functions of only one position
More informationChapter 4. Motion in Two Dimensions
Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail Will treat projectile motion and uniform circular motion
More informationNewton s first law. Projectile Motion. Newton s First Law. Newton s First Law
Newton s first law Projectile Motion Reading Supplemental Textbook Material Chapter 13 Pages 88-95 An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same
More informationv 1 parabolic orbit v 3 m 2 m 3
Example 10.5 Exploding Projectile An instrument-carrying projectile of mass m 1 accidentally explodes at the top of its trajectory. The horizontal distance between launch point and the explosion is. The
More informationFirst Year Physics: Prelims CP1. Classical Mechanics: Prof. Neville Harnew. Problem Set III : Projectiles, rocket motion and motion in E & B fields
HT017 First Year Physics: Prelims CP1 Classical Mechanics: Prof Neville Harnew Problem Set III : Projectiles, rocket motion and motion in E & B fields Questions 1-10 are standard examples Questions 11-1
More informationCHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS
CHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS OBJECTIVES After studying the material of this chapter, the student should be able to: represent the magnitude and direction of a vector using a protractor
More informationProjectile Motion and 2-D Dynamics
Projectile Motion and 2-D Dynamics Vector Notation Vectors vs. Scalars In Physics 11, you learned the difference between vectors and scalars. A vector is a quantity that includes both direction and magnitude
More informationMATH 32A: MIDTERM 1 REVIEW. 1. Vectors. v v = 1 22
MATH 3A: MIDTERM 1 REVIEW JOE HUGHES 1. Let v = 3,, 3. a. Find e v. Solution: v = 9 + 4 + 9 =, so 1. Vectors e v = 1 v v = 1 3,, 3 b. Find the vectors parallel to v which lie on the sphere of radius two
More informationChapter 2. Kinematics in One Dimension. continued
Chapter 2 Kinematics in One Dimension continued 2.6 Freely Falling Bodies Example 10 A Falling Stone A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement
More informationKinematics in Two Dimensions; Vectors
Kinematics in Two Dimensions; Vectors Vectors & Scalars!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.!! Vectors
More informationAnalytical Mechanics - Extra Problems
Analytical Mechanics - Extra Problems Physics 105, F17 (R) are review problems. Review problems are those that have already been covered in prior courses, mostly Intro to Physics I and II. Some are math
More informationVARIABLE MASS PROBLEMS
VARIABLE MASS PROBLEMS Question 1 (**) A rocket is moving vertically upwards relative to the surface of the earth. The motion takes place close to the surface of the earth and it is assumed that g is the
More informationChapter 8 Solutions. The change in potential energy as it moves from A to B is. The change in potential energy in going from A to B is
Chapter 8 Solutions *8. (a) With our choice for the zero level for potential energy at point B, U B = 0. At point A, the potential energy is given by U A = mgy where y is the vertical height above zero
More informationChapter 2 One-Dimensional Kinematics. Copyright 2010 Pearson Education, Inc.
Chapter 2 One-Dimensional Kinematics Units of Chapter 2 Position, Distance, and Displacement Average Speed and Velocity Instantaneous Velocity Acceleration Motion with Constant Acceleration Applications
More informationMotion in Two Dimensions
P U Z Z L E R This airplane is used by NASA for astronaut training. When it flies along a certain curved path, anything inside the plane that is not strapped down begins to float. What causes this strange
More informationExam 2. May 21, 2008, 8:00am
PHYSICS 101: Fundamentals of Physics Exam 2 Exam 2 Name TA/ Section # May 21, 2008, 8:00am Recitation Time You have 1 hour to complete the exam. Please answer all questions clearly and completely, and
More informationChapter 4. Motion in Two Dimensions. Professor Wa el Salah
Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail. Will treat projectile motion and uniform circular
More informationPhysics Mid-Term Practice Exam
Physics Mid-Term Practice Exam Multiple Choice. Identify the choice that best completes the statement or answers the question. 1. Which one of the following problems would NOT be a part of physics? a.
More informationKINETICS: MOTION ON A STRAIGHT LINE. VELOCITY, ACCELERATION. FREELY FALLING BODIES
014.08.06. KINETICS: MOTION ON A STRAIGHT LINE. VELOCITY, ACCELERATION. FREELY FALLING BODIES www.biofizika.aok.pte.hu Premedical course 04.08.014. Fluids Kinematics Dynamics MECHANICS Velocity and acceleration
More informationNormal Force. W = mg cos(θ) Normal force F N = mg cos(θ) F N
Normal Force W = mg cos(θ) Normal force F N = mg cos(θ) Note there is no weight force parallel/down the include. The car is not pressing on anything causing a force in that direction. If there were a person
More informationMotion in 2- and 3-dimensions. Examples: non-linear motion (circles, planetary orbits, etc.) flight of projectiles (shells, golf balls, etc.
Motion in 2- and 3-dimensions Examples: HPTER 3 MOTION IN TWO & THREE DIMENSIONS General properties of vectors the displacement vector position and velocity vectors acceleration vector equations of motion
More informationOptical Physics of Rifle Scopes
Optical Physics of Rifle Scopes A Senior Project By Ryan Perry Advisor, Dr. Glen Gillen Department of Physics, California Polytechnic University SLO June 8, 207 Approval Page Title: Optical Analysis of
More informationProjectile motion. Solution: Observation:
Projectile motion by Frank Owen, PhD, P.E., Alpha Omega Engineering, Inc., September 014 Let s explore projectile motion a bit more. For the following analysis, ignore air resistance and any variations
More informationConstants: Acceleration due to gravity = 9.81 m/s 2
Constants: Acceleration due to gravity = 9.81 m/s 2 PROBLEMS: 1. In an experiment, it is found that the time t required for an object to travel a distance x is given by the equation = where is the acceleration
More informationConstants: Acceleration due to gravity = 9.81 m/s 2
Constants: Acceleration due to gravity = 9.81 m/s 2 PROBLEMS: 1. In an experiment, it is found that the time t required for an object to travel a distance x is given by the equation = where is the acceleration
More informationFormative Assessment: Uniform Acceleration
Formative Assessment: Uniform Acceleration Name 1) A truck on a straight road starts from rest and accelerates at 3.0 m/s 2 until it reaches a speed of 24 m/s. Then the truck travels for 20 s at constant
More informationChapter Review USING KEY TERMS UNDERSTANDING KEY IDEAS. Skills Worksheet. Multiple Choice
Skills Worksheet Chapter Review USING KEY TERMS Complete each of the following sentences by choosing the correct term from the word bank. mass gravity friction weight speed velocity net force newton 1.
More informationMarble Launch Experiment
Marble Launch Experiment Purpose The intent of this experiment is to numerically trace the path of a marble launched into the air at an angle in order to observe the parabolic nature of the trajectory.
More informationMotion in a 2 and 3 dimensions Ch 4 HRW
Motion in a and 3 dimensions Ch 4 HRW Motion in a plane D Motion in space 3D Projectile motion Position and Displacement Vectors A position vector r extends from a reference point (usually the origin O)
More informationChapter 3 Acceleration
Chapter 3 Acceleration Slide 3-1 Chapter 3: Acceleration Chapter Goal: To extend the description of motion in one dimension to include changes in velocity. This type of motion is called acceleration. Slide
More informationPhysics 121. Tuesday, January 29, 2008.
Physics 121. Tuesday, January 29, 2008. This is where your instructor grew up. Schiphol (Amsterdam Airport) = cemetery of ships. Physics 121. Tuesday, January 29, 2008. Topics: Course announcements Quiz
More informationVector and Relative motion discussion/ in class notes. Projectile Motion discussion and launch angle problem. Finish 2 d motion and review for test
AP Physics 1 Unit 2: 2 Dimensional Kinematics Name: Date In Class Homework to completed that evening (before coming to next class period) 9/6 Tue (B) 9/7 Wed (C) 1D Kinematics Test Unit 2 Video 1: Vectors
More informationPhysics Fall Mechanics, Thermodynamics, Waves, Fluids. Recap: Position and displacement
Physics 5 Fall 28 Mechanics, Thermodynamics, Waves, Fluids Lecture 3: motion in a straight line II Slide 3- Recap: Position and displacement In one dimension, position can be described by a positive or
More informationJames T. Shipman Jerry D. Wilson Charles A. Higgins, Jr. Omar Torres. Chapter 2 Motion Cengage Learning
James T. Shipman Jerry D. Wilson Charles A. Higgins, Jr. Omar Torres Chapter 2 Motion Defining Motion Motion is a continuous change in position can be described by measuring the rate of change of position
More informationMOTION IN A PLANE. Chapter Four MCQ I. (a) 45 (b) 90 (c) 45 (d) 180
Chapter Four MOTION IN A PLANE MCQ I 4.1 The angle between A = ˆi + ˆj and B = ˆi ˆj is (a) 45 (b) 90 (c) 45 (d) 180 4.2 Which one of the following statements is true? (a) A scalar quantity is the one
More informationVectors and Coordinate Systems
Vectors and Coordinate Systems In Newtonian mechanics, we want to understand how material bodies interact with each other and how this affects their motion through space. In order to be able to make quantitative
More informationChapter 4. Motion in Two Dimensions. With modifications by Pinkney
Chapter 4 Motion in Two Dimensions With modifications by Pinkney Kinematics in Two Dimensions covers: the vector nature of position, velocity and acceleration in greater detail projectile motion a special
More informationComponents of a Vector
Vectors (Ch. 1) A vector is a quantity that has a magnitude and a direction. Examples: velocity, displacement, force, acceleration, momentum Examples of scalars: speed, temperature, mass, length, time.
More informationQ3.1. A. 100 m B. 200 m C. 600 m D m E. zero. 500 m. 400 m. 300 m Pearson Education, Inc.
Q3.1 P 400 m Q A bicyclist starts at point P and travels around a triangular path that takes her through points Q and R before returning to point P. What is the magnitude of her net displacement for the
More informationAn Overview of Mechanics
An Overview of Mechanics Mechanics: The study of how bodies react to forces acting on them. Statics: The study of bodies in equilibrium. Dynamics: 1. Kinematics concerned with the geometric aspects of
More informationProblem Set. Assignment #1. Math 3350, Spring Feb. 6, 2004 ANSWERS
Problem Set Assignment #1 Math 3350, Spring 2004 Feb. 6, 2004 ANSWERS i Problem 1. [Section 1.4, Problem 4] A rocket is shot straight up. During the initial stages of flight is has acceleration 7t m /s
More informationGraphical Vector Addition
Vectors Chapter 4 Vectors and Scalars Measured quantities can be of two types Scalar quantities: only require magnitude (and proper unit) for description. Examples: distance, speed, mass, temperature,
More informationRandom sample problems
UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 125 / LeClair Spring 2009 Random sample problems 1. The position of a particle in meters can be described by x = 10t 2.5t 2, where t is in seconds.
More informationMotion in Two and Three Dimensions
PH 1-1D Spring 013 Motion in Two and Three Dimensions Lectures 5,6,7 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
More informationb) (6) How far down the road did the car travel during the acceleration?
General Physics I Quiz 2 - Ch. 2-1D Kinematics June 17, 2009 Name: For full credit, make your work clear to the grader. Show the formulas you use, all the essential steps, and results with correct units
More informationMAT 272 Test 1 Review. 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same
11.1 Vectors in the Plane 1. Let P = (1,1) and Q = (2,3). Find the unit vector u that has the same direction as. QP a. u =< 1, 2 > b. u =< 1 5, 2 5 > c. u =< 1, 2 > d. u =< 1 5, 2 5 > 2. If u has magnitude
More informationPHYS 1111L - Introductory Physics Laboratory I
PHYS 1111L - Introductory Physics Laboratory I Laboratory Advanced Sheet Projectile Motion Laboratory 1. Objective. The objective of this laboratory is to predict the range of a projectile set in motion
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationQuiz Number 3 PHYSICS March 11, 2009
Instructions Write your name, student ID and name of your TA instructor clearly on all sheets and fill your name and student ID on the bubble sheet. Solve all multiple choice questions. No penalty is given
More informationVocabulary Preview. Oct 21 9:53 AM. Projectile Motion. An object shot through the air is called a projectile.
Projectile Trajectory Range Launch angle Vocabulary Preview Projectile Motion Projectile Motion An object shot through the air is called a projectile. A projectile can be a football, a bullet, or a drop
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics Module 10 - Lecture 24 Kinematics of a particle moving on a curve Today,
More informationAntiderivatives. Definition A function, F, is said to be an antiderivative of a function, f, on an interval, I, if. F x f x for all x I.
Antiderivatives Definition A function, F, is said to be an antiderivative of a function, f, on an interval, I, if F x f x for all x I. Theorem If F is an antiderivative of f on I, then every function of
More informationProjectile Motion trajectory Projectile motion
Projectile Motion The path that a moving object follows is called its trajectory. An object thrown horizontally is accelerated downward under the influence of gravity. Gravitational acceleration is only
More information2D Motion Projectile Motion
2D Motion Projectile Motion Lana heridan De Anza College Oct 3, 2017 Last time vectors vector operations Warm Up: Quick review of Vector Expressions Let a, b, and c be (non-null) vectors. Let l, m, and
More informationChapter 2 Kinematics in One Dimension
Chapter 2 Kinematics in One Dimension The Cheetah: A cat that is built for speed. Its strength and agility allow it to sustain a top speed of over 100 km/h. Such speeds can only be maintained for about
More information