( ) in the following way. ( ) < 2

Save this PDF as:

Size: px
Start display at page:

Download "( ) in the following way. ( ) < 2"

Transcription

1 Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer Verical rockes objecs subjeced o graviy As an obbec moves is posiion is a funcion of ime. For is posiion funcion we will denoe he variable s( ). For insance when s!! in seconds s( ) we are being old wha posiion on he horizonal or verical number line he paricle occupies a differen values of.!! Example 1) For s show is posiion on he number line for d =1 = U = = ( S( S S S1 1 ( 4 When an obbec moves is posiion changes over ime. So we can say ha he velociy funcion v of he posiion funcion over ime. We know his o be a derivaive and can hus say ha v For convenience sake we will define v Moion ( ) > ( ) in he following way. ( ) < v ( ) = v v Horizonal Line obbec moves o he righ obbec moves o he lef obbec sopped Verical Line obbec moves up obbec moves down obbec sopped ( ) is he change s-( ). Speed is no synonymous wih velociy. Speed does no indicae direcion. So we define he speed funcion: ->..<! v!. The speed of an obbec mus eiher be posiive or zero (meaning ha he obbec is sopped). = ( ) The definiion of acceleraion is he change of velociy over ime. We know his o be a derivaive and can hus say ha a( ) = v-( ) = s--( ). So given he posiion funcion s( ) we can now deermine boh he velociy and acceleraion funcion. On your cars you have wo devices o change he velociy: acceleraor brake Le us hink as somehing acceleraing he obbec o be some exernal force like wind or curren. For convenience sake le us define he acceleraion funcion like his: Moion ( ) > a( ) < a( ) = a Horizonal Line obbec acceleraing o he righ obbec acceleraing o he lef velociy no changing Verical Line obbec acceleraing upwards obbec acceleraing downwards velociy no changing Jus because an obbec s acceleraion is zero does no mean ha he obbec is sopped. I means ha he velociy is no changing. Wha device do you have on your cars ha keeps he car s acceleraion equal o zero_ cruise conrol Also bus because you have a posiive acceleraion does no mean ha you are moving o he righ. For insance suppose you were walking o he righ v when all of a sudden a large wind sared o blow o he lef [ ( ) < ] a [ ( ) > ]. Wha would ha do o your velociy_ slow you down. AB Soluions Su Schwarz

2 ! + Example ) Given ha a paricle is moving along a horizonal line wih posiion funcion s (. (. The velociy funcion v! and he acceleraion funcion a Le s complee he char for he firs 5 seconds and show where he obbec is on he number line. s( ) v ( )! v( )! a ( ) Descripion of he paricle s moion -4 4 moving lef acceleraing o he righ moving lef bu slower sill acceleraing o he righ - sopped sill acceleraing o he righ 3-1 moving o he righ acceleraing o he righ moving o he righ faser sill acceleraing o he righ moving o he righ faser ye sill acceleraing o he righ I is oo much work o do such work for complicaed funcions. We are generally ineresed when he paricle is sopped or when i has no acceleraion. We are also ineresed when he obbec is speeding up or slowing down. Realizing ha an obbec s velociy is eiher posiive (moving righ) negaive (moving lef) or zero (sopped) and an obbec s acceleraion is eiher posiive negaive or zero (consan speed) we can now use a char o deermine all he possibiliies of an obbec s moion as if you were looking a i from above. a( ) > a( ) < a( ) = ( ) > speeding up slowing down consan velociy righ ( ) < slowing down speeding up consan velociy lef sopped acceleraing righ sopped acceleraing lef sopped no acceleraion v v v! + Example 3) A paricle is moving along a horizonal line wih posiion funcion s 3 4. Do an analysis! of he paricle s direcion (righ lef) acceleraion moion (speeding up slowing down) & posiion. Sep 1: v! So v 3 Sep : Make a number line of v a d3 g h ( ) showing when v( ) i jjjjjjjjjjj he obbec is sopped and he sign and 3 direcion of he obbec a imes o he lef and righ of ha. Assume h. Sep 3: a( ) =. Does a( ) = _ No Sep 4: Make a number line of a( ) showing a( ) ijjjjjjjjjjjjjjjjjjjjjj when he obbec has a posiive and negaive acceleraion. Scale i exacly like he v( ) number line. slowing down speeding up Sep 5: Make a moion line direcly below moion i jjjjjjjjjjj he las wo puing all criical values 3 muliplying he signs and inerpreing h according o he char above. g Sep 6: Make a posiion graph o show where he posiion llllld3lllllllllldlllllllll obbec is a criical imes and how i moves AB Soluions Su Schwarz

3 ! + + Example 3) A paricle is moving along a horizonal line wih posiion funcion s 8 ( (. Do an! analysis of he paricle s direcion acceleraion moion (speeding up or slowing down) and posiion. v 1 ( v 3 v ( = U = ( a( ) = 3! 1 =!! + = (! + ) = (! )(! ) = = v( ) a( ) SSSSSSSSSSSSSSSSSSSSSSSSS !( SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS I9F!<9F\$->..<!E>!-I9F!<9F\$->..<!E> 79*#9\$ SSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSS !( *= *=( *= >9-#*#9\$ (!( Noe ha he posiion graph is no like he oher hree graphs. I simply shows he posiion he obbec has wih respec o he origin and criical imes of is movemen found by seing v ( ) and a( ) d.! + + ( ) is measured in fee v is he iniial velociy (velociy a d ) and s is he iniial posiion!( if s( ) is measured in meers. When an obbec is subbeced o graviy is posiion funcion is given by s 13 v s where is measured in seconds s (posiion a d ). The formula is given by s v s! + +! + v o and he acceleraion funcion a( ) =!. This is he acceleraion due o graviy on earh. When an obbec is hrown upward i is subbeced o graviy We are usually ineresed how high he paricle reaches and how fas i is going when i impacs he ground or waer. Le us analyze wha hese mean: From our original s 13 v s we can calculae he velociy funcion v When an obbec reaches is maximum heigh When an obbec his he ground wha is is wha is is velociy_ v = final posiion_ s = So o find he maximum heigh of an obbec se v So o find he velociy of an obbec when i solve for and find s( ) his he ground se s( ) d solve for and find v( ) Example 4). A probecile is launched verically upward from ground level wih an iniial velociy of 11 f/sec. a. Find he velociy and speed b. How high will he probecile c. Find he speed of he a d 3 and d 5 seconds. rise_ probecile when i his he ground.! + s v( ) =! + 11! v 13!)*%-./->..< = 13!)*%-./ v 4 (!)*%-./!->..<! = (!)*%-./! v( ) =! + 11 = = 11 = 54! ( ) + ( ) s s = 183!)*! + = s 13 11! 13! 6 = 6! ( ) + v v 6 11!)*%-./ AB Soluions Su Schwarz

4 Example 5) The equaions for free fall a he surfaces of Mars Earh and Jupier (s in meers in seconds) are: Mars: s 15 3 Earh: s ( 58 Jupier: s! 11 5 ((. How long would i ake a rock iniially a res in a space capsule over he plane o reach a velociy of 16.6 m/sec_ Mars Earh Jupier v = 1353 v = 1353 = ( 5 (3!-./ = 1538(!-./ = 563!-./ v 5 5 = 1353 Example 6) A rock hrown verically upward from he surface of he moon a a velociy of 4m/sec reaches a heigh of s = (! 5 meers in seconds. a) Find he rock s velociy and acceleraion as a b) How long did i ake he rock o reach is highes funcion of ime. (The acceleraion in his case poin_ is he acceleraion on he moon)!! v a 1. 6! = ) = v ( !-./ c) How high did he rock go_ d) How long did i ake he rock o reach half is maximum heigh_ ( )! ( ) s (! 5 = 8 s m 5! ( + 8 = ) = ( 5 8!-./ e) How long was he rock alof_ e) Find he rock s speed when hiing he moon.! = s ( 5! 51 = U =!-./ v( 3) = 4! 1. 6( 3)! v 4 speed = 4 m/sec Example 7) A ball is dropped from he op of he Washingon Monumen which is 555 fee high. a) How long will i ake for he ball o hi he b) Find he ball s speed a impac. ground_ s( ) =! = 13 = 444 ) = 458!-./! f / sec + v mph Example 8) Paul has bough a icke on a special roller coaser a an amusemen park which moves in a sraigh line. The posiion s U! 1 ( ) of he car in fee afer seconds is given by: s( ) =! +.. a) Find he velociy and acceleraion of he b) When is he roller coaser sopped_ roller coaser afer seconds_! +! + v 5 5( a 53! 5(! 5 + 5(! = / = U =!-./! c) When is Paul speeding up and slowing down_ d) Where is Paul a criical imes of his ride_! ( = ) = ( ( ) ( ) Q>..<!E>! U( U U 1!QI9F!<9F\$!((U ) = s = = 8 s = 56 = 1 s = AB Soluions Su Schwarz

5 Sraigh Line Moion - Homework A paricle is moving along a horizonal line wih posiion funcion as given. Do an analysis of he paricle s direcion acceleraion moion (speeding up or slowing down) and posiion. +!! = ) =! 1. s 3 v 3 a! +!! + ) = U! ) =. s 3 8 ( v a 3 1 v( )!a ( ) -I9F#\$J!<9F\$!->..<#\$J!E> 79*#9\$ SSSSSSSSSSSSSSSSSSSSSSSSSS s( ) SSSSSSSSSSSSSSSSSSSSSSSS SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS *= *= 11 v( )!a ( ) s( ) !SSSSSSSSSSSSSSSSSSSSSSSSS! SSSSSSSSSSSSSSSSSSSSSSSSS! I9F!<9F\$!->..<!E>-I9F!<9F\$->..<!E> 79*#9\$ SSSSSSSSSSS! !SSSSSSSSSSS! *= *= *=1 S( 8 3. s( ) =! + 8! ( s v! + 1! ( ) = ( a ( ) =! ) 8 U = v( ) =! ) = a( ) =! ( 1) ( + 1) v( ) SSSSSSSSSSSSSS! !SSSSSSSSSSSSSS v( ) SSSSSSSSSSSSSSSSSSSSSS (!a ( ) -I9F!<9F\$!->..<!E>!-I9F!<9F\$->..<!E> SSSSSSSSSSSSS!++++++SSSSSSSSSSSS *#9\$ ( s( ) SSSSSSSSSSSSSSSSSSSSSSSS *= S18 S14 *=( 1 *=!a( ) s( ) I9F#\$J!<9F\$!->..<#\$J!E> 79*#9\$ SSSSSSSSSSSSSSSSSSSSSSS A 45-caliber bulle fired sraigh up from he surface of he moon would reach a heigh of s =! 53 fee afer seconds. On Earh in he absence of air is heigh would be! s =! 13 fee afer seconds. How long would i ake he bulle o hi he ground in eiher case_ *= 3 1 *= Earh s (! ) = 16 = 83 ) = 5 sec Moon s (! ) =. 6 = 83 ) = 3 sec AB Soluions Su Schwarz

6 6. A ball fired downward from a heigh of 11 fee his he ground in seconds. Find is iniial velociy.! + o + = s 13 v 11! 3( + v + 11 = ) v =!( o vo =!(!)*%-./ 7. A probecile is fired verically upward (earh) from ground level wih an iniial velociy of 16 f/sec. a. How long will i ake for he probecile o hi b. How high will he probecile ge_ he ground_ o! + = s 13 13! 13! 1 = 1!-./ v( ) =! + 13 = = 13 ) = 54! ( ) + s (!)* 8. A helicoper pilo drops a package when he helicoper is f. above he ground rising a f/sec. a. How long will i ake for he package o hi b. Wha is he speed of he package a impac_ he ground_! + + =!!! s 16 ( ) ) = s sec v( ) =! 3 + v! ( ) + = f/sec 9. A man drops a quarer from a bridge. How high is he bridge if he quarer his he waer 4 seconds laer_! + = s 13 s s( ) =! 13 (() + s = s o = 43!)* 1. A probecile fired upward from ground level is o reach a maximum heigh of 16 fee. Wha is is iniial velociy_ s 16 v 16 v 3 v v = 3! 16 + ( 3) = = 16 = 1 sec v! + =! + = f/sec = AB Soluions Su Schwarz

7 11. A probecile is fired verically upward wih an iniial velociy of 96 f/sec from a ower 56 fee high. a. How long will i ake for he probecile o reach b. Wha is is maximum heigh_ is maximum heigh.! + +! + = ) = s v 83!-./! + +! + = ) = s v 83!-./ s (!)* c. How long will i ake he probecile o reach d. Wha is he velociy when i passes he saring is saring heigh on he way down_ poin on he way down_! + + = ) = s ! 13! 3 3!-./ 83!)*%-./ e. How long will i ake o hi he ground_ f. Wha will be is speed when i impacs he ground_! + + = s ) =! 16! 6! 16 8 sec v( 8) =! 3( 8) + 96 = 16 f / sec 1. John s car runs ou of gas as i goes up a hill. The car rolls o a sop hen sars rolling backwards. As i rolls is displacemen d +! ( ) in fee from he boom of he hill a seconds since he car ran ou of gas is given by: d a. When is his velociy posiive_ Wha does his b. When did he car sar o roll backwards_ mean in real world erms_ How far was i from he boom of he hill a ha ime_ v( ) = 31! >. < sec - going up he hill v( ) = 31! < sec - d f c. If John keeps his foo off he brake when will d. How far was John from he boom of he hill he be a he boom of he hill_ when he ran ou of gas_ 1 f ! = ) + ( 531!-./ d Ray is a sky-diver. When he free-falls his downward velociy v from he ime of he bump is given by: v calculaor for he firs 3 seconds of his dive.! ( ) ( ) fee per second is a funcion of seconds ( ) and a( ) on your measured in f/sec. Plo v a. Wha is Ray s acceleraion when he firs bumps_ b. Wha appears o be he erminal velociy I#7 v Why does he acceleraion decrease over ime_ /1 ( ) _! f /sec - air resisance 41!)*%-./ AB Soluions Su Schwarz

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

More information

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

More information

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

More information

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,

More information

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

More information

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

More information

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

More information

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

More information

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..

More information

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

More information

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

More information

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

More information

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:

More information

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

More information

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

More information

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

More information

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

More information

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.

More information

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)

More information

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

More information

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

More information

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

More information

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

More information

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

More information

4.2 Trigonometric Functions; The Unit Circle

4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.

More information

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

More information

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

More information

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

More information

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

More information

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.

More information

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

More information

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

More information

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

More information

EXERCISES AND PROBLEMS

Exercises and Problems 71 EXERCISES AND PROBLEMS The icon in fron of a problem indicaes ha he problem can be done on a Dnamics Workshee. Dnamics Workshees are found a he back of he Suden Workbook. If ou

More information

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

More information

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

More information

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

More information

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.

More information

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.

More information

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

More information

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

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

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

More information

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

More information

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

More information

AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

More information

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

More information

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

More information

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,

More information

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

More information

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

More information

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.

More information

Chapter 4: Exponential and Logarithmic Functions

Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

More information

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

More information

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

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS

EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67 - FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1 - RANIEN Uni conen 2 Undersand he ransien behaviour of resisor-capacior (R) and resisor-inducor (RL) D circuis

More information

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)

More information

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

More information

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

More information

Physics 111 Fall 2007 Electric Currents and DC Circuits

Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels

More information

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

More information

Present Value Methodology

Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

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

More information

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.

More information

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

More information

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

More information

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

More information

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

More information

Improper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].

Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,

More information

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

More information

Mr. Kepple. Motion at Constant Acceleration 1D Kinematics HW#5. Name: Date: Period: (b) Distance traveled. (a) Acceleration.

Moion Consn Accelerion 1D Kinemics HW#5 Mr. Kepple Nme: De: Period: 1. A cr cceleres from 1 m/s o 1 m/s in 6.0 s. () Wh ws is ccelerion? (b) How fr did i rel in his ime? Assume consn ccelerion. () Accelerion

More information

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.

More information

Solution of a differential equation of the second order by the method of NIGAM

Tire : Résoluion d'une équaion différenielle du second[...] Dae : 16/02/2011 Page : 1/6 Soluion of a differenial equaion of he second order by he mehod of NIGAM Summarized: We presen in his documen, a

More information

Stochastic Optimal Control Problem for Life Insurance

Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

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

More information

Topic Overview. Learning Objectives. Capital Budgeting Steps: WHAT IS CAPITAL BUDGETING?

Chaper 10: THE BASICS OF CAPITAL BUDGETING Should we build his plan? Topic Overview Projec Types Capial Budgeing Decision Crieria Payback Period Discouned Payback Period Ne Presen Value () Inernal Rae

More information

Morningstar Investor Return

Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

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

More information

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

More information

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

More information

Module 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur

Module 3 - & -C Transiens esson 0 Sudy of DC ransiens in - and -C circuis Objecives Definiion of inducance and coninuiy condiion for inducors. To undersand he rise or fall of curren in a simple series

More information

Equities: Positions and Portfolio Returns

Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi

More information

Signal Processing and Linear Systems I

Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information

Astable multivibrator using the 555 IC.(10)

Visi hp://elecronicsclub.cjb.ne for more resources THE 555 IC TIMER The 555 IC TIMER.(2) Monosable mulivibraor using he 555 IC imer...() Design Example 1 wih Mulisim 2001 ools and graphs..(8) Lile descripion

More information

Signal Rectification

9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

More information

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

Pulse-Width Modulation Inverters

SECTION 3.6 INVERTERS 189 Pulse-Widh Modulaion Inverers Pulse-widh modulaion is he process of modifying he widh of he pulses in a pulse rain in direc proporion o a small conrol signal; he greaer he conrol

More information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

More information

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.

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information

Stability. Coefficients may change over time. Evolution of the economy Policy changes

Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,

More information

HFCC Math Lab Intermediate Algebra - 13 SOLVING RATE-TIME-DISTANCE PROBLEMS

HFCC Mah Lab Inemeiae Algeba - 3 SOLVING RATE-TIME-DISTANCE PROBLEMS The vaiables involve in a moion poblem ae isance (), ae (), an ime (). These vaiables ae elae by he equaion, which can be solve fo any

More information

C Fast-Dealing Property Trading Game C

AGES 8+ C Fas-Dealing Propery Trading Game C Y Collecor s Ediion Original MONOPOLY Game Rules plus Special Rules for his Ediion. CONTENTS Game board, 6 Collecible okens, 28 Tile Deed cards, 16 Wha he Deuce?

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

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

More information

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

More information

C Fast-Dealing Property Trading Game C

If you are already an experienced MONOPOLY dealer and wan a faser game, ry he rules on he back page! AGES 8+ C Fas-Dealing Propery Trading Game C Y Original MONOPOLY Game Rules plus Special Rules for his

More information

Chapter 6 Interest Rates and Bond Valuation

Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Long-erm deb-loosely, bonds wih a mauriy of one year or more Shor-erm deb-less han a year o mauriy, also called unfunded deb Bond-sricly

More information

Variance Swap. by Fabrice Douglas Rouah

Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach

More information