Chapter 3: vectors and 2d motion. Summing vectors graphically

Transcription

1 Chapte 3: ectos and 2d motion Summing ectos gaphically Tiangula method Resultant ecto 1

2 Summing ectos gaphically Paallelogam method Resultant ecto Subtacting ectos gaphically 2

3 Components of a ecto A A x A y A A x y Acosθ Asinθ Components of ectos What ae the components of the displacement ecto of supeman? 3

4 Popeties of ectos Equality: two ectos ae equal if they hae the same magnitude and diection Addition: tiangula method; paallelogam method o adding components (note that A+BB+A) Subtaction: A-BA+(-B) A + B ( Ax + Bx, Ay + By ) Multiplying o diiding by a scala: 3A is a ecto with the same diection as A and thee times the magnitude. 3A (3A x,3ay ) Stategy with ectos Select a coodinate system Sketch the ecto to be added o subtact and label them Find the x and y component of each ecto Find the esultant components Use the Pythagoean theoem to find the magnitude of the esultant ecto Use a suitable tigonometic function to find the angle the esultant ecto makes with the positie x axis 4

5 Vectos: example A hike begins a tip by fist walking 25.0 km southeast fom he base camp. On the second day she walks 40.0 km in the diection 60.0 degees notheast, at which point she discoes a foest ange s towe. a) Detemine the components of the hike s displacement on the 1 st and 2 nd day b) Detemine the total displacement Example of summing ectos What is the magnitude and diection of the esultant foce applied to the donkey? 5

6 6 2-D motion 2-D motion: definitions t Aeage elocity displacement f i t t lim0 Instantaneous elocity Aeage acceleation Instantaneous acceleation t a t a t lim0

7 2-D motion: questions A body cannot hae acceleation if its speed is constant a) Tue b) False A paticle can hae constant elocity and aying speed a) Tue b) False Thee may be acceleation in a ca when you ca is on cuise contol a) Tue b) False Thee is acceleation in you ca when you hit the bakes a) Tue b) False Pojectile motion 7

8 Pojectile motion It can be descibed as a supeposition of two independent motions in the x and y diections Poided ai esistance is negligible, the hoizontal component of elocity x emains constant The etical component of the acceleation is equal to the fee fall acceleation g The etical component of the elocity y and the displacement in the y diection ae identical to those of a feely falling body Pojectile motion: stategy Select a coodinate system and sketch the path of the paticle Resole the initial elocity in its x and y components Teat the hoizontal and etical motion independently Follow method fo constant elocity to analyse x motion Follow method fo constant acceleation to analyse the y motion 8

9 An Alaskan escue plane dops a package of emegency ations to standed hikes. The plane is taeling hoizontally at 40.0 m/s at height of 100 m aboe the gound. a) Whee does the package stike the gound elatie to the point at which it was eleased? b) What ae the hoizontal and etical components of the elocity of the package just befoe it hits the gound? Pojectile motion: example Pojectile motion As a pojectile moes in its paabolic path, the elocity and acceleation ectos ae pependicula to each othe: a) Eeywhee along its path b) At the peak of its path c) Nowhee along its path d) Not enough infomation If you ae caying a ball and unning at constant speed and wish to thow the ball so that you can catch it when it comes back down, should you: a) Thow the ball at an angle of 45deg and maintain the same speed? b) Thow the ball staight in the ai, and slow down to catch it? c) Thow the ball staight in the ai and maintain the same speed? d) Not enough infomation 9

10 Pojectile motion: example A basketball playe 2.0 m tall, wants to make a basket fom a distance of 10.0 m. If he shoots the ball at 45.0 deg, at what initial speed must he thow the ball so that it goes though the hoop without stiking the backboad? Relatie elocity A passenge at the ea of a tain taeling at 15 m/s elatie to the Eath thows a baseball with a speed 15 m/s in the diection opposite to the motion of the tain. What is the elocity of the baseball elatie to the Eath? ball eath ball tain + tain eath 15 m/s 15 m/s 10

11 Relatie elocity The boat is heading due noth as it cosses a wide ie with elocity of 10.0 km/h elatie to the wate. The ie has a unifom elocity of 5.00 km/h due east. Detemine the elocity of the boat with espect to an obsee on the iebank? boat eath boat ie + ie eath Relatie elocity If now the boat moes at the same speed 10 km/h elatie to the wate but wants to tael due noth, in what diection should it moe elatie to the wate? boat eath boat ie + ie eath 11

12 Motion on an incline A cate moes down along an incline, stating fom est at a height of 50 cm. The incline makes 30 deg with the hoizontal axis. Suppose thee is no fiction between the cate and the incline. a) Detemine the acceleation along the diection of the motion b) How does the acceleation ay with time? c) How does the elocity ay with time? d) How does the position ay with time? (poide equation and gaph) Angula displacement θ s Cicula motion Peiod (T) time fo one otation Acceleation in cicula motion a 2 12