# Relative velocity in one dimension

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Connexions module: m Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies peraining o moion are characerisically relaive in naure. The measuremens, describing moion, are subjec o he sae of moion of he frame of reference wih respec o which measuremens are made. Our day o day percepion of moion is generally earh's view a view common o all bodies a res wih respec o earh. However, we encouner occasions when here is percepible change o our earh's view. One such occasion is raveling on he ciy rains. We nd ha i akes lo longer o overake anoher rain on a parallel rack. Also, we see wo people alking while driving separae cars in he parallel lane, as if hey were saionary o each oher!. In erms of kinemaics, as a maer of fac, hey are acually saionary o each oher - even hough each of hem are in moion wih respec o ground. In his module, we se ourselves o sudy moion from a perspecive oher han ha of earh. Only condiion we subjec ourselves is ha wo references or wo observers making he measuremens of moion of an objec, are moving a consan velociy (We shall learn aferward ha wo such reference sysems moving wih consan velociy is known as inerial frames, where Newon's laws of moion are valid.). The observers hemselves are no acceleraed. There is, however, no resricion on he moion of he objec iself, which he observers are going o observe from dieren reference sysems. The moion of he objec can very well be acceleraed. Furher, we shall sudy relaive moion for wo caegories of moion : (i) one dimension (in his module) and (ii) wo dimensions (in anoher module). We shall skip hree dimensional moion hough wo dimensional sudy can easily be exended o hree dimensional moion as well. 1 Relaive moion in one dimension We sar here wih relaive moion in one dimension. I means ha he individual moions of he objec and observers are along a sraigh line wih only wo possible direcions of moion. 1.1 Posiion of he poin objec We consider wo observers A and B. The observer A is a res wih earh, whereas observer B moves wih a velociy v BA wih respec o he observer A. The wo observers wach he moion of he poin like objec C. The moions of B and C are along he same sraigh line. noe: I helps o have a convenion abou wriing subscriped symbol such as v BA. The rs subscrip indicaes he eniy possessing he aribue (here velociy) and second subscrip indicaes he eniy wih respec o which measuremen is made. A velociy like v BA shall, herefore, mean velociy of B wih respec o A. Version 1.8: Oc 3, :46 am GMT-5 hp://creaivecommons.org/licenses/by/2.0/

2 Connexions module: m The posiion of he objec C as measured by he wo observers A and B are x CA and x CB as shown in he gure. The observers are represened by heir respecive frame of reference in he gure. Posiion Figure 1 Here, x CA = x BA + x CB 1.2 Velociy of he poin objec We can obain velociy of he objec by diereniaing is posiion wih respec o ime. As he measuremens of posiion in wo references are dieren, i is expeced ha velociies in wo references are dieren, because one observer is a res, whereas oher observer is moving wih consan velociy. and v CA = x CA v CB = x CB Now, we can obain relaion beween hese wo velociies, using he relaion x CA = x BA + x CB and diereniaing he erms of he equaion wih respec o ime : x CA = x BA + x CB

3 Connexions module: m v CA = v BA + v CB Relaive velociy Figure 2 The meaning of he subscriped velociies are : v CA : velociy of objec "C" wih respec o "A" v CB : velociy of objec "C" wih respec o "B" v BA : velociy of objec "B" wih respec o "A" Example 1 Problem : Two cars, sanding a disance apar, sar moving owards each oher wih speeds 1 m/s and 2 m/s along a sraigh road. Wha is he speed wih which hey approach each oher? Soluion : Le us consider ha "A" denoes Earh, "B" denoes rs car and "C" denoes second car. The equaion of relaive velociy for his case is : v CA = v BA + v CB Here, we need o x a reference direcion o assign sign o he velociies as hey are moving opposie o each oher and should have opposie signs. Le us consider ha he direcion of he velociy of B is in he reference direcion, hen

4 Connexions module: m Relaive velociy Figure 3 Now : v BA = 1 m/s and v CA = 2 m/s. v CA = v BA + v CB 2 = 1 + v CB v CB = 2 1 = 3 m/s This means ha he car "C" is approaching "B" wih a speed of -3 m/s along he sraigh road. Equivalenly, i means ha he car "B" is approaching "C" wih a speed of 3 m/s along he sraigh road. We, herefore, say ha he wo cars approach each oher wih a relaive speed of 3 m/s. 1.3 Acceleraion of he poin objec If he objec being observed is acceleraed, hen is acceleraion is obained by he ime derivaive of velociy. Diereniaing equaion of relaive velociy, we have : v CA = v BA + v CB v CA = v BA + v CB The meaning of he subscriped acceleraions are : a CA = a BA + a CB a CA : acceleraion of objec "C" wih respec o "A" a CB : acceleraion of objec "C" wih respec o "B"

5 Connexions module: m a BA : acceleraion of objec "B" wih respec o "A" Bu we have resriced ourselves o reference sysems which are moving a consan velociy. This means ha relaive velociy of "B" wih respec o "A" is a consan. In oher words, he acceleraion of "B" wih respec o "A" is zero i.e. a BA = 0. Hence, a CA = a CB The observers moving a consan velociy, herefore, measure same acceleraion of he objec. As a maer of fac, his resul is characerisics of inerial frame of reference. The reference frames, which measure same acceleraion of an objec, are inerial frames of reference. 2 Inerpreaion of he equaion of relaive velociy The imporan aspec of relaive velociy in one dimension is ha velociy has only wo possible direcions. We need no use vecor noaion o wrie or evaluae equaion of relaive velociies in one dimension. The velociy, herefore, can be reaed as signed scalar variable; plus sign indicaing velociy in he reference direcion and minus sign indicaing velociy in opposie o he reference direcion. 2.1 Equaion wih reference o earh The equaion of relaive velociies refers velociies in relaion o dieren reference sysem. v CA = v BA + v CB We noe ha wo of he velociies are referred o A. In case, A denoes Earh's reference, hen we can convenienly drop he reference. A velociy wihou reference o any frame shall hen mean Earh's frame of reference. v C = v B + v CB v CB = v C v B This is an imporan relaion. This is he working relaion for relaive moion in one dimension. We shall be using his form of equaion mos of he ime, while working wih problems in relaive moion. This equaion can be used eecively o deermine relaive velociy of wo moving objecs wih uniform velociies (C and B), when heir velociies in Earh's reference are known. Le us work ou an exercise, using new noaion and see he ease of working. Example 2 Problem : Two cars, iniially 100 m disan apar, sar moving owards each oher wih speeds 1 m/s and 2 m/s along a sraigh road. When would hey mee? Soluion : The relaive velociy of wo cars (say 1 and 2) is : v 21 = v 2 v 1 Le us consider ha he direcion v 1 is he posiive reference direcion. Here, v 1 = 1 m/s and v 2 = -2 m/s. Thus, relaive velociy of wo cars (of 2 w.r. 1) is : v 21 = 2 1 = 3 m / s This means ha car "2" is approaching car "1" wih a speed of -3 m/s along he sraigh road. Similarly, car "1" is approaching car "2" wih a speed of 3 m/s along he sraigh road. Therefore, we can say ha wo cars are approaching a a speed of 3 m/s. Now, le he wo cars mee afer ime :

6 Connexions module: m = Displacemen Relaive velociy = = 33.3 s 2.2 Order of subscrip There is sligh possibiliy of misundersanding or confusion as a resul of he order of subscrip in he equaion. However, if we observe he subscrip in he equaion, i is easy o formulae a rule as far as wriing subscrip in he equaion for relaive moion is concerned. For any wo subscrips say A and B, he relaive velociy of A (rs subscrip) wih respec o B (second subscrip) is equal o velociy of A (rs subscrip) subraced by he velociy of B (second subscrip) : v AB = v A v B and he relaive velociy of B (rs subscrip) wih respec o A (second subscrip) is equal o velociy of B (rs subscrip) subraced by he velociy of A (second subscrip): v BA = v B v A 2.3 Evaluaing relaive velociy by making reference objec saionary An inspecion of he equaion of relaive velociy poins o an ineresing feaure of he equaion. We need o emphasize ha he equaion of relaive velociy is essenially a vecor equaion. In one dimensional moion, we have aken he libery o wrie hem as scalar equaion : v BA = v B v A Now, he equaion comprises of wo vecor quaniies ( v B and v A ) on he righ hand side of he equaion. The vecor v A is acually he negaive vecor i.e. a vecor equal in magniude, bu opposie in direcion o v A. Thus, we can evaluae relaive velociy as following : 1. Apply velociy of he reference objec (say objec "A") o boh objecs and render he reference objec a res. 2. The resulan velociy of he oher objec ("B") is equal o relaive velociy of "B" wih respec o "A". This concep of rendering he reference objec saionary is explained in he gure below. In order o deermine relaive velociy of car "B" wih reference o car "A", we apply velociy vecor of car "A" o boh cars. The relaive velociy of car "B" wih respec o car "A" is equal o he resulan velociy of car "B".

7 Connexions module: m Relaive velociy Figure 4 This echnique is a very useful ool for consideraion of relaive moion in wo dimensions. 2.4 Direcion of relaive velociies For a pair of wo moving objecs moving uniformly, here are wo values of relaive velociy corresponding o wo reference frames. The values dier only in sign no in magniude. This is clear from he example here. Example 3 Problem : Two cars sar moving away from each oher wih speeds 1 m/s and 2 m/s along a sraigh road. Wha are relaive velociies? Discuss he signicance of heir sign. Soluion : Le he cars be denoed by subscrips 1 and 2. Le us also consider ha he direcion v 2 is he posiive reference direcion, hen relaive velociies are :

8 Connexions module: m Relaive velociy Figure 5 v 12 = v 1 v 2 = 1 2 = 3 m / s v 21 = v 2 v 1 = 2 ( 1 ) = 3 m / s The sign aached o relaive velociy indicaes he direcion of relaive velociy wih respec o reference direcion. The direcions of relaive velociy are dieren, depending on he reference objec. However, wo relaive velociies wih dieren direcions mean same physical siuaion. Le us read he negaive value rs. I means ha car 1 moves away from car 2 a a speed of 3 m/s in he direcion opposie o ha of car 2. This is exacly he physical siuaion. Now for posiive value of relaive velociy, he value reads as car 2 moves from car 1 in he direcion of is own velociy. This also is exacly he physical siuaion. There is no conradicion as far as physical inerpreaion is concerned. Imporanly, he magniude of approach whaever be he sign of relaive velociy is same.

9 Connexions module: m Relaive velociy Figure Relaive velociy.vs. dierence in velociies I is very imporan o undersand ha relaive velociy refers o wo moving bodies no a single body. Also ha relaive velociy is a dieren concep han he concep of "dierence of wo velociies", which may perain o he same or dieren objecs. The dierence in velociies represens dierence of nal velociy and iniial velociy and is independen of any order of subscrip. In he case of relaive velociy, he order of subscrips are imporan. The expression for wo conceps viz relaive velociy and dierence in velociies may look similar, bu hey are dieren conceps. 2.6 Relaive acceleraion We had resriced ou discussion up o his poin for objecs, which moved wih consan velociy. The quesion, now, is wheher we can exend he concep of relaive velociy o acceleraion as well. The answer is yes. We can aach similar meaning o mos of he quaniies - scalar and vecor boh. I all depends on aaching physical meaning o he relaive concep wih respec o a paricular quaniy. For example, we measure poenial energy (a scalar quaniy) wih respec o an assumed daum. Exending concep of relaive velociy o acceleraion is done wih he resricion ha measuremens of individual acceleraions are made from he same reference. If wo objecs are moving wih dieren acceleraions in one dimension, hen he relaive acceleraion is equal o he ne acceleraion following he same working relaion as ha for relaive velociy. For example, le us consider han an objec designaed as "1" moves wih acceleraion "a 1 " and he oher objec designaed

10 Connexions module: m as "2" moves wih acceleraion " a 2 " along a sraigh line. Then, relaive acceleraion of "1" wih respec o "2" is given by : a 12 = a 1 a 2 Similarly,relaive acceleraion of "2" wih respec o "1" is given by : a 21 = a 2 a 1 3 Worked ou problems Example 4: Relaive moion Problem :Two rains are running on parallel sraigh racks in he same direcion. The rain, moving wih he speed of 30 m/s overakes he rain ahead, which is moving wih he speed of 20 m/s. If he rain lenghs are 200 m each, hen nd he ime elapsed and he ground disance covered by he rains during overake. Soluion : Firs rain, moving wih he speed of 30 m/s overakes he second rain, moving wih he speed of 20 m/s. The relaive speed wih which rs rain overakes he second rain, v 12 = v 1 v 2 = = 10 m/s. The gure here shows he iniial siuaion, when faser rain begins o overake and he nal siuaion, when faser rain goes pas he slower rain. The oal disance o be covered is equal o he sum of each lengh of he rains (L1 + L2) i.e = 400 m. Thus, ime aken o overake is :

11 Connexions module: m The oal relaive disance Figure 7: rain. The oal relaive disance o cover during overake is equal o he sum of lenghs of each = = 40 s. In his ime inerval, he wo rains cover he ground disance given by: s = 30 x x 40 = = 2000 m. Exercise 1 (Soluion on p. 12.) In he quesion given in he example, if he rains ravel in he opposie direcion, hen nd he ime elapsed and he ground disance covered by he rains during he period in which hey cross each oher. 4 Check your undersanding Check he module iled Relaive velociy in one dimension (Check your undersanding) 1 o es your undersanding of he opics covered in his module. 1 "Relaive velociy in one dimension(applicaion)" <hp://cnx.org/conen/m14035/laes/>

12 Connexions module: m Soluions o Exercises in his Module Soluion o Exercise 1 (p. 11) v 12 = v 1 v 2 = 30 ( 20 ) = 50 m/s. The oal disance o be covered is equal o he sum of each lengh of he rains i.e = 400 m. Thus, ime aken o overake is : = = 8 s. Now, in his ime inerval, he wo rains cover he ground disance given by: s = 30 x x 8 = = 400 m. In his case, we nd ha he sum of he lenghs of he rains is equal o he ground disance covered by he rains, while crossing each oher.

### Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

### Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

### Chapter 2 Kinematics in One Dimension

Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

### Newton s Laws of Motion

Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

### Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

### Acceleration Lab Teacher s Guide

Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

### Rotational Inertia of a Point Mass

Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

### Chapter 7. Response of First-Order RL and RC Circuits

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

### MOTION ALONG A STRAIGHT LINE

Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,

### A Curriculum Module for AP Calculus BC Curriculum Module

Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.

### Motion Along a Straight Line

Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

### Imagine a Source (S) of sound waves that emits waves having frequency f and therefore

heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing

### Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

### AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

### AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

### Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

### Section 7.1 Angles and Their Measure

Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

### Inductance and Transient Circuits

Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

### Permutations and Combinations

Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

### The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

### cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

### CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

### 4. International Parity Conditions

4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

### Kinematics in 1-D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.

Chaper 2 Kinemaics in 1-D From Problems and Soluions in Inroducory Mechanics (Draf ersion, Augus 2014) Daid Morin, morin@physics.harard.edu As menioned in he preface, his book should no be hough of as

### m m m m m correct

Version 055 Miderm 1 OConnor (05141) 1 This prin-ou should have 36 quesions. Muliple-choice quesions ma coninue on he ne column or pae find all choices before answerin. V1:1, V:1, V3:3, V4:, V5:1. 001

### Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009.

Name: Teacher: DO NOT OPEN THE EXMINTION PPER UNTIL YOU RE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINL EXMINTION June 2009 Value: 100% General Insrucions This examinaion consiss of wo pars. Boh pars

### 11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

### A Mathematical Description of MOSFET Behavior

10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

### Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

### Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

### Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

### Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

### Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

### 1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

### RC (Resistor-Capacitor) Circuits. AP Physics C

(Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

### Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

### Chapter 2: Principles of steady-state converter analysis

Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

### 4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

### The Derivative of a Constant is Zero

Sme Simple Algrihms fr Calculaing Derivaives The Derivaive f a Cnsan is Zer Suppse we are l ha x x where x is a cnsan an x represens he psiin f an bjec n a sraigh line pah, in her wrs, he isance ha he

### 9. Capacitor and Resistor Circuits

ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

### ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

### USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were

### WHAT ARE OPTION CONTRACTS?

WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

### SOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3.

SOLI MEHNIS TUTORIL GER SYSTEMS This work covers elemens of he syllabus for he Edexcel module 21722P HN/ Mechanical Principles OUTOME 3. On compleion of his shor uorial you should be able o do he following.

### Part 1: White Noise and Moving Average Models

Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

### Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

### Cointegration: The Engle and Granger approach

Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

### MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM

MA6-A Calculus III 006 Fall Homework 4 Soluions Due 9/9/006 00AM 97 #4 Describe in words he surface 3 A half-lane in he osiive x and y erriory (See Figure in Page 67) 97 # Idenify he surface cos We see

### Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is \$613.

Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

### AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

### MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

### Section 5.1 The Unit Circle

Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P

### Capital budgeting techniques

Capial budgeing echniques A reading prepared by Pamela Peerson Drake O U T L I N E 1. Inroducion 2. Evaluaion echniques 3. Comparing echniques 4. Capial budgeing in pracice 5. Summary 1. Inroducion The

### Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

### RC, RL and RLC circuits

Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

### 17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

### The Torsion of Thin, Open Sections

EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

### Circuit Types. () i( t) ( )

Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

### Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

### Chabot College Physics Lab RC Circuits Scott Hildreth

Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

### Chapter 8: Regression with Lagged Explanatory Variables

Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

### AP Physics Velocity and Linear Acceleration Unit 1 Problems:

Uni 1 Problems: Linear Velociy and Acceleraion This enire se of problems is due he day of he es. I will no accep hese for a lae grade. * = Problems we do ogeher; all oher problems are homework (bu we will

### THE PRESSURE DERIVATIVE

Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.

### A Probability Density Function for Google s stocks

A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural

### Using RCtime to Measure Resistance

Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a buil-in ADC (Analog o Digial Converer)

### Chapter 1.6 Financial Management

Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

### Double Entry System of Accounting

CHAPTER 2 Double Enry Sysem of Accouning Sysem of Accouning \ The following are he main sysem of accouning for recording he business ransacions: (a) Cash Sysem of Accouning. (b) Mercanile or Accrual Sysem

### Form measurement systems from Hommel-Etamic Geometrical tolerancing in practice DKD-K-02401. Precision is our business.

Form measuremen sysems from Hommel-Eamic Geomerical olerancing in pracice DKD-K-02401 Precision is our business. Drawing enries Tolerance frame 0.01 0.01 Daum leer Tolerance value in mm Symbol for he oleranced

### RC Circuit and Time Constant

ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

### Differential Equations and Linear Superposition

Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

### 6.5. Modelling Exercises. Introduction. Prerequisites. Learning Outcomes

Modelling Exercises 6.5 Inroducion This Secion provides examples and asks employing exponenial funcions and logarihmic funcions, such as growh and decay models which are imporan hroughou science and engineering.

### Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

### 1 HALF-LIFE EQUATIONS

R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

### Technical Appendix to Risk, Return, and Dividends

Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### Two Compartment Body Model and V d Terms by Jeff Stark

Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

### 5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()

### Transient Analysis of First Order RC and RL circuits

Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage

### CHAPTER FIVE. Solutions for Section 5.1

CHAPTER FIVE 5. SOLUTIONS 87 Soluions for Secion 5.. (a) The velociy is 3 miles/hour for he firs hours, 4 miles/hour for he ne / hour, and miles/hour for he las 4 hours. The enire rip lass + / + 4 = 6.5

### Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

### Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets

Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have

### Steps for D.C Analysis of MOSFET Circuits

10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.

### SELF-EVALUATION FOR VIDEO TRACKING SYSTEMS

SELF-EVALUATION FOR VIDEO TRACKING SYSTEMS Hao Wu and Qinfen Zheng Cenre for Auomaion Research Dep. of Elecrical and Compuer Engineering Universiy of Maryland, College Park, MD-20742 {wh2003, qinfen}@cfar.umd.edu

### The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

### Strategic Optimization of a Transportation Distribution Network

Sraegic Opimizaion of a Transporaion Disribuion Nework K. John Sophabmixay, Sco J. Mason, Manuel D. Rossei Deparmen of Indusrial Engineering Universiy of Arkansas 4207 Bell Engineering Cener Fayeeville,

### A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

### Density Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).

FW 662 Densiy-dependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Long-erm

### DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR

Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios

### Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

### 3 Runge-Kutta Methods

3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

### Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds

OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030-364X eissn 1526-5463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:

### The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

### The Kinetics of the Stock Markets

Asia Pacific Managemen Review (00) 7(1), 1-4 The Kineics of he Sock Markes Hsinan Hsu * and Bin-Juin Lin ** (received July 001; revision received Ocober 001;acceped November 001) This paper applies he

### Period 4 Activity Solutions: Transfer of Thermal Energy

Period 4 Aciviy Soluions: Transfer of Thermal nergy 4.1 How Does Temperaure Differ from Thermal nergy? a) Temperaure Your insrucor will demonsrae molecular moion a differen emperaures. 1) Wha happens o

### Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

### Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach

Opimal Consumpion and Insurance: A Coninuous-Time Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems

### House Price Index (HPI)

House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales

### Individual Health Insurance April 30, 2008 Pages 167-170

Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve