Ground Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions

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PC11 Fndamentals of Physis I Letres 7 and 8 Motion in Two Dimensions A/Prof Tay Sen Chan 1 Grond Rles Swith off yor handphone and paer Swith off yor laptop ompter and keep it No talkin while letre is

1 PC11 Fndamentals of Physis I Letres 7 and 8 Motion in Two Dimensions A/Prof Tay Sen Chan 1 Grond Rles Swith off yor handphone and paer Swith off yor laptop ompter and keep it No talkin while letre is oin on No ossipin while the letre is oin on Raise yor hand if yo hae qestion to ask Be on time for letre Be on time to ome bak from the reess break to ontine the letre Brin yor letrenotes to letre Position and Displaement Aerae Veloity The position of an objet is desribed by its position etor, r The displaement of the objet is defined as the hane in its position Δr = r f - r i In two- or threedimensional kinematis, eerythin is the same as in one-dimensional motion exept that we mst now se fll etor notation Plane View 3 The aerae eloity is the ratio of the displaement to the time interal for the displaement r t The diretion of the aerae eloity is the diretion of the displaement etor, Δr The aerae eloity between points is independent of the path taken It is dependent on the displaement Plane View 4

2 Instantaneos Veloity The instantaneos eloity is the limit of the aerae eloity as Δt approahes zero = lim t 0 r dr t dt The diretion of small displaement (hane in positions) tells the diretion that the partile is headin at the moment. 5 Instantaneos Veloity, ont The diretion of the instantaneos eloity etor at any point in a partile s path is alon a line tanent to the path at that point and in the diretion of motion The manitde of the instantaneos eloity etor is the speed The speed is a salar qantity 6 Aerae Aeleration Aerae Aeleration, ont The aerae aeleration of a partile as it moes is defined as the hane in the instantaneos eloity etor diided by the time interal drin whih that hane ors. f i a t t t f i As a partile moes, Δ an be fond in different ways The aerae aeleration is a etor qantity direted alon Δ Plane View 7 8

Instantaneos Aeleration The instantaneos aeleration is the limit of the aerae aeleration as Δt approahes zero a lim = t 0 t d dt The diretion of small hane in eloities tells the diretion of

3 Instantaneos Aeleration The instantaneos aeleration is the limit of the aerae aeleration as Δt approahes zero a lim = t 0 t d dt The diretion of small hane in eloities tells the diretion of aeleration. x 9 Prodin An Aeleration Varios hanes in a partile s motion may prode an aeleration, sh as: The manitde of the eloity etor may hane The diretion of the eloity etor may hane Een if the manitde remains onstant Both may also hane simltaneosly 10 Kinemati Eqations for Two- Dimensional Motion When the two-dimensional motion has a onstant aeleration, a series of eqations an be deeloped that desribe the motion These eqations will be similar to those of one-dimensional kinematis Kinemati Eqations, Position etor Veloity r xˆi yˆj dr ˆ ˆ xiyj dt Sine aeleration is onstant, we an also find an expression for the eloity as a fntion of time: f = i + at 11 1

Kinemati Eqations, 3 The eloity etor an be represented by its omponents f is enerally not alon the diretion of either i or at Kinemati Eqations, 4 The position etor an also be expressed as a fntion

= i + at 13 14 Kinemati Eqations, 5 Kinemati Eqations, Components The etor representation of the position etor r f is enerally not in the same diretion as i or as a i r f and f are enerally not in

4 Kinemati Eqations, 3 The eloity etor an be represented by its omponents f is enerally not alon the diretion of either i or at Kinemati Eqations, 4 The position etor an also be expressed as a fntion of time: r f = r i + i t + ½ at This indiates that the position etor is the sm of three other etors: The initial position etor r i The displaement resltin from i t The displaement resltin from ½ at f = i + at Kinemati Eqations, 5 Kinemati Eqations, Components The etor representation of the position etor r f is enerally not in the same diretion as i or as a i r f and f are enerally not in the same diretion The eqations for final eloity and final position are etor eqations, therefore they may also be written in omponent form This shows that two-dimensional motion at onstant aeleration is eqialent to two independent motions One motion in the x-diretion and r f = r i + i t + ½ at the other in the y-diretion 15 16

5 Kinemati Eqations, Component Eqations Projetile Motion f = i + at beomes xf = xi + a x t and yf = yi + a y t r f = r i + i t + ½ at beomes x f = x i + xi t + ½ a x t and y f = y i + yi t + ½ a y t 17 An objet may moe in both x and y diretions simltaneosly The form of twodimensional motion we will deal with is alled projetile motion Front View 18 Assmptions of Projetile Motion The free-fall aeleration is onstant oer the rane of motion is direted downward The effet of air frition is neliible With these assmptions, an objet in projetile motion will follow a paraboli path This path is alled the trajetory Verifyin the Paraboli Trajetory Referene frame hosen y is ertial with pward positie Aeleration omponents a y = - and a x = 0 Initial eloity omponents xi = i os and yi = i sin i Θ 19 0

Verifyin the Paraboli Trajetory, ont Rane and Maximm Heiht of a Projetile Displaements x f = xi t = ( i os t y f = yi t + ½a y t = ( i sin t -½t Combinin the eqations and remoin t ies: y tan x x i i

6 Verifyin the Paraboli Trajetory, ont Rane and Maximm Heiht of a Projetile Displaements x f = xi t = ( i os t y f = yi t + ½a y t = ( i sin t -½t Combinin the eqations and remoin t ies: y tan x x i i os i This is in the form of y = ax bx whih is the standard form of a parabola 1 When analyzin projetile motion, two harateristis are of speial interest The rane, R, is the horizontal distane of the projetile The maximm heiht the projetile reahes is h Heiht of a Projetile, eqation The maximm heiht of the projetile an be fond in terms of the initial eloity etor: h i sin i This eqation is alid only for symmetri motion Rane of a Projetile, eqation The rane of a projetile an be expressed in terms of the initial eloity etor: i sin R i This is alid only for symmetri trajetory 3 We will derie them. 4

7 Projetile Formlation Let the initial eloity be. Let the anle sbsripted by and x-axis be θ. Alon the x-diretion, x os t x t os Alon the Y-diretion, y 1 sin t t 1 Sbstitte (1) into (): y θ x Maximm ors when the y-omponent of the eloity is eqal to 0, i.e. the objet is at the moment of omin down. sin t 0 sin t 0 sin t Sbstitte (3) into (): sin y h sin sin sin h h sin or i 1 3 sin sin i h To derie the maximm rane (R) alon the x-diretion, we let y = 0 in eqation. 1 y sin t t R The distane traelled in x-diretion Why? 1 sin t t t sin t sin sin t t 0 0 sin (A+B) = sin A os B + os A sin B 4 for y t x x x sin seonds os t sin os R sin os sin is : What is the maximm ale of R, and the orrespondin? Sine sin 1, sin 1, i.e., 90 When 45 R R R R is a maximm when sin 90, sin 45 45

8 Rane of a Projetile, final The maximm rane ors at i = 45 o Complementary anles will prode the same rane Bt the maximm heiht will be different for the two anles The times of the fliht will be different for the two anles Projetile Motion Problem Solin Hints Selet a oordinate system Resole the initial eloity into x and y omponents Analyze the horizontal motion sin onstant eloity tehniqes Analyze the ertial motion sin onstant aeleration tehniqes Remember that both diretions share the same time 9 30 Non-Symmetri Projetile Motion Follow the eneral rles for projetile motion Break the y-diretion into parts p and down or symmetrial bak to initial heiht and then the rest of the heiht May be non-symmetri in other ways Example. One stratey in a snowball fiht is to throw a snowball at a hih anle oer leel rond. Then, while yor opponent is wathin that snowball, yo throw a seond one at a low anle timed to arrie before or at the same time as the first one. Assme that both snowballs are thrown with a speed of 5.0 m/s. The first is thrown at an anle of 70.0 with respet to the horizontal. The release point of snowball and the hit point of the opponent are at the same leel. (a) At what anle shold the seond snowball be thrown to arrie at the same point as the first? (b) How many seonds later shold the seond snowball be thrown after the first in order for both to arrie at the same time? 31 3

9 We hae deried (b) R sin Sine R,, are onstant, sin will also be a onstant. Let the soltion of be, where 90. We hae, or 180. So,, or 90 R Answer: (a) The seond snowball shold be thrown at Uniform Cirlar Motion Chanin Veloity in Uniform Cirlar Motion Uniform irlar motion ors when an objet moes in a irlar path with a onstant speed An aeleration exists sine the diretion of the motion is hanin This hane in eloity is related to an aeleration The eloity etor is always tanent to the path of the objet Plane View The hane in the eloity etor is de to the hane in diretion The etor diaram shows = f - i 35 36

Centripetal Aeleration Centripetal Aeleration, ont The aeleration is always perpendilar to the path of the motion The aeleration always points toward the enter of the irle of motion This aeleration

10 Centripetal Aeleration Centripetal Aeleration, ont The aeleration is always perpendilar to the path of the motion The aeleration always points toward the enter of the irle of motion This aeleration is alled the entripetal aeleration 37 The manitde of the entripetal aeleration etor is ien by a C r By the se of same ratio, this an be deried as follows: Δ Δr = r Δ = Δr r Δ Δr = x Δt r Δt a C r 38 Period The period, T, is the time reqired for one omplete reoltion (one omplete yle) As one omplete yle is the distane of the irmferene. The speed of the partile wold be the irmferene of the irle of motion diided by the period. Speed = distane/time, so time = distane/speed r Therefore, the period is T = 39 Example. The astronat orbitin the Earth in Fire is preparin to dok with a Westar VI satellite. The satellite is in a irlar orbit 600 km aboe the Earth's srfae, where the free-fall aeleration is 8.1 m/s. Take the radis of the Earth as 6400 km. Determine the speed of the satellite and the time interal reqired to omplete one orbit arond the Earth. 40

11 so 8.1 m/s Tanential Aeleration and Speed of satellite in order to maintain the orbit Time to omplete 1 yle of orbit 41 The manitde of the eloity old also be hanin In this ase, there wold be a tanential aeleration Obsere what happens at the enter (my hand) when I make the ball to rotate faster Cirlar motion on my hand 4 Total Aeleration The tanential aeleration ases the hane in the speed of the partile The radial aeleration omes from a hane in the diretion of the eloity etor. (In the textbook by Serway, the diretion or radial aeleration is away from enter not important.) a = a t + a r a a t a r 43 Total Aeleration, eqations d The tanential aeleration: a t = dt The radial aeleration: a r = a = The total aeleration: Manitde a a a r t Why the strin does not slak and the ball does not fly towards the enter? r 44

Relatie Veloity Two obserers moin relatie to eah other enerally do not aree on the otome of an experiment For example, obserers A and B below see different paths for the ball Relatie Veloity,

12 Relatie Veloity Two obserers moin relatie to eah other enerally do not aree on the otome of an experiment For example, obserers A and B below see different paths for the ball Relatie Veloity, eneralized Referene frame S is stationary Referene frame S is moin at 0 This also means that S moes at 0 relatie to S Define time t = 0 as that time when the oriins oinide Relatie Veloity, eqations Aeleration in Different Frames of Referene The positions as seen from the two referene frames (r is the position seen from stationary frame, r seen from moin frame) are related throh the eloity r = r 0 t The deriatie of the position eqation will ie the eloity eqation = 0 These are alled the Galilean transformation eqations (d1-9) 47 The deriatie of the eloity eqation will ie the aeleration eqation The aeleration of the partile measred by an obserer in one frame of referene is the same as that measred by any other obserer moin at a onstant eloity relatie to the first frame. Why? r = r 0 t 0 is a onstant, so 0 has no effet on the = 0 hane in eloity, ie, a = a. 48

13 Example. Two swimmers, Tom and Jerry, start toether at the same point on the bank of a wide stream that flows with a speed. Both moe at the same speed ( > ), relatie to the water. Tom swims downstream a distane L and then pstream the same distane. Jerry swims so that his motion relatie to the Earth is perpendilar to the banks of the stream. He swims the distane L and then bak the same distane, so that both swimmers retrn to the startin point. Whih swimmer retrns first? Answer: For Tom, his speed downstream is +, while his speed pstream is L L Therefore, the total time for Tom is t1 1 L. 49 For Jerry, his ross-stream speed (both ways) is Ths, the total time for Jerry is t L L L t1 1 L L 1 Sine 1 1, t1 t, or Jerry, who swims ross-stream, retrns first. L L L t Example. A Coast Gard tter detets an nidentified ship at a distane of 0.0 km in the diretion 15.0 east of north. The ship is traelin at 6.0 km/h on a orse at 40.0 east of north. The Coast Gard wishes to send a speedboat to interept the essel and inestiate it. If the speedboat traels 50.0 km/h, in what diretion shold it head? Express the diretion as a ompass bearin with respet to de north. Answer: Choose the x-axis alon the 0-km distane. The y-omponents of the displaements of the ship and the speedboat mst aree: 6 km htsin km htsin sin The speedboat shold head eastofnorth 6 km/h 50 km/h 51 Example. How lon does it take an atomobile traelin in the left lane at 60.0 km/h to pll alonside a ar traelin in the riht lane at 40.0 km/h if the ars' front bmpers are initially 100 m apart? (Disk ) Answer: The bmpers are initially 100 m apart (0.1 km). After time t the bmper of the leadin ar traels 40.0t km, while the bmper of the hasin ar traels 60.0t km. Sine the ars are side by side at time t we hae t = 60t t = 0.1/0 = hr = x 3600 se = 18 s 60 t 60 km/h 40 km/h 0.1 km 40 t 5

Examples. Constant speed (A) Horizontal Trak Examples. This time the ar will be plled alon the horizontal trak by a strin and a weiht hanin oer a plley, and will be aeleratin onstantly.

14 Examples. Constant speed (A) Horizontal Trak Examples. This time the ar will be plled alon the horizontal trak by a strin and a weiht hanin oer a plley, and will be aeleratin onstantly. The ball is fired at point X. Where will the ball land this time? (The horizontal trak is lon enoh for the ar not to fall off when the ball lands.) Point X (C) The ar is plled alon by a strin and a weiht hanin oer a plley. (B) Tilted Trak 53 54

Ground Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions

Ground Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions PC11 Fndamentals of Physics I Lectres 7 and 8 Motion in Two Dimensions A/Prof Tay Sen Chan 1 Grond Rles Switch off yor handphone and paer Switch off yor laptop compter and keep it No talkin while lectre