INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example, say an aeoplane flies fom Heathow to Pais If the ai is still (no wind), the pilot can simply head the plane towads its destination If the wind is blowing fom west to east, the pilot must use vectos to detemine the coect heading Thus a vecto is a mathematically impotant tool in physics 2. Vectos in Physics 1
SCALARS VERSUS VECTORS A scala is a numbe with units it can eithe be positive, negative o zeo (volume, tempeatue, time) When asking fo diections, the answe you ae given is 5km nothwest this infomation contains both the distance (magnitude) and diection A vecto is a mathematical quantity with both a diection and a magnitude The answe given above is the displacement vecto Othe examples of vecto quantities ae the velocity and acceleation of an object The magnitude of a velocity vecto is its speed, and its diection is the diection of motion A vecto is witten in boldface with a small aow above it, i.e. Magnitudes ae given in italics, i.e. = 5km 2. Vectos in Physics 2
PYTHAGORAS THEOREM: REMINDER Pythagoas Theoem is essential when solving poblems with vectos Given the magnitude and diection of a vecto, it can be used to find the x and y components A x = Acosθ and A y = Asinθ Given the components of a vecto, its magnitude and diection can be found A = (A x2 + A y2 ) and θ = tan -1 (A y /A x ) 2. Vectos in Physics 3
THE COMPONENTS OF A VECTOR (1) When giving city diections, such as 5km nothwest, it would be impossible to walk in a staight line City steets ae often aanged in noth-south and eastwest diections Walking in a combination of these diections will eventually lead to you destination In the above figue, instead of walking in a staight line nothwest, you walk a distance west, then a distance noth, you aive at you destination What you will have done is esolved the displacement vecto into east-west and noth south components It is necessay to set up a coodinate system fo a 2D system x and y positive diections, with oigin O In 3D, we would have to include a z axis 2. Vectos in Physics 4
THE COMPONENTS OF A VECTOR (2) Suppose an ant leaves its nest at the oigin and afte some time it is at the location given by the vecto The vecto has a magnitude = 1.5m and points in the diection θ = 25 above x axis extends a distance x in the x diection, and a distance y in the y diection The quantities x and y ae the scala components of the vecto The scala components can be found by using standad tigonometic elations x = cos 25.0 = 1.5 0.906 = 1.36m Likewise y = sin 25.0 = 0.634m Thus the ant s final displacement is equivalent to what it would be if it had walked 1.36m in the x, then 0.634m in the y diection = ( x2 + y2 ) = (1.36 2 + 0.634 2 ) = 1.5m θ = sin -1 (0.634/1.5) = cos -1 (1.36/1.5) = tan -1 (0.634/1.36) = 25.0 2. Vectos in Physics 5
THE COMPONENTS OF A VECTOR: EXAMPLES An exploe wants to find the height of a cliff. He stands with his back to the base of the cliff, then maches staight away fom it fo 500 feet. At this point he lies on the gound and measues the angle fom the hoizontal to the top of the cliff. If the angle is 34, how high is the cliff? What is the staight line distance fom the exploe to the top of the cliff? Find A x and A y fo the vecto with magnitude and diection given by A = 3.5m and θ = 66 espectively. Find B and θ fo the vecto with components B x = 75.5m and B y = 6.2m. B A 2. Vectos in Physics 6
THE COMPONENTS OF A VECTOR: SIGNS To detemine the signs of a vecto s components, it is only necessay to obseve the diection in which they point The vecto A has magnitude 7.25m. Find its components fo diection angles of 5.0, 125, 245 and 335. 2. Vectos in Physics 7
CALCULATORS AND THE DIRECTION ANGLE You may need to add 180 when using a calculato to detemine the diection angle If A x = -0.5m and A y = 1.0m, you calculato will give the following esult θ = tan -1 (1.0/-0.5) = -63, which doesn t coespond to the specified vecto Sketching A shows that the angle should be between 90 and 180 So to obtain the coect angle, add 180 to the esult Example: B has components B x = -2.1m and B y = - 1.7m. Find the diection vecto θ fo this vecto. Finally some tigonometic identities A x = A sinθʹ = A sin(90 -θ) = A cosθ A y = A cosθʹ = A cos(90 -θ) = A sinθ 2. Vectos in Physics 8
ADDING VECTORS GRAPHICALLY (1) If you ae given diections fom the oigin to mach 5 paces noth, then 3 paces east, and if these two displacements ae epesented by the vectos and B A, then the total displacement fom the oigin to the destination is given by the vecto C Thus C is the vecto sum of A and B To add the vectos A and B gaphically, place the tail of B at the head of A. The sum C = A + B is the vecto extending fom the tail of A to the head of B A vecto is defined by its magnitude and diection, egadless of its location 2. Vectos in Physics 9
ADDING VECTORS GRAPHICALLY (2) The sum of the vectos is independent of the ode in which the vectos ae added C = A + B = B + A If A is 5.0m, and B is 4.0m with the espective angles as shown, then C can be found gaphically using a ule and potacto 2. Vectos in Physics 10
ADDING VECTORS USING COMPONENTS C x = A x + B x and C y + A y + B y If A = 5.0m and B = 4.0m, with the same angles as the pevious slide, then A x = 5.0 cos(60) = 2.5m, A y = 5.0 sin(60) = 4.33m B x = 4.0 cos(20) = 3.76m, B y = 4.0 sin(20) = 1.37m C x = A x + B x = 2.5 + 3.76 = 6.26m C y = A y + B y = 4.33 + 1.37 = 5.70m Magnitude C = (C x2 + C y2 ) = (6.26 2 + 5.70 2 ) = 8.47m Angle θ = tan -1 (C y /C x ) = tan -1 (5.70/6.26) = 42.3 2. Vectos in Physics 11
SUBTRACTING VECTORS Finding D A B is equivalent to = D = A + ( B) The negative of a vecto is epesented by an aow of the same length as the oiginal vecto, but pointing in the opposite diection. That is, multiplying a vecto by -1 eveses its diection To find D = A B using the components method: D x = A x B x and D y = A y B y And the magnitude and diection of D ae found in the usual way Example: Fo the vectos given in the pevious slide, find the components of D = A B. Also find D and θ. 2. Vectos in Physics 12
UNIT VECTORS Unit vectos povide a convenient way of expessing an abitay vecto in tems of its components xˆ The unit vectos and ae defined to be the dimensionless vectos of unit magnitude pointing in the positive x and y diections xˆ The figue below shows and on a 2D coodinate system Since unit vectos have no physical dimensions, such as mass, length o time, they ae used to specify diection only ŷ ŷ 2. Vectos in Physics 13
MULTIPLYING UNIT VECTORS BY SCALARS To see the utility of unit vectos, conside the effect of multiplying a vecto by a scala Multiplying a vecto by 3 inceases its magnitude but does not change its diection Multiplying by -3 inceases its magnitude and eveses its diection If a vecto A has scala components A x and A y then we can wite it in tems of its vecto components A = A xˆ yˆ x + Ay C = A + B = ( A x + Bx ) xˆ + ( Ay + By )yˆ D = A B = ( Ax Bx ) xˆ + ( Ay By )yˆ Unit vectos povide a useful way of keeping tack of the x and y components 2. Vectos in Physics 14
POSITION VECTORS Imagine a 2D coodinate system as shown Position is indicated by a vecto fom the oigin to the location in question The position vecto is efeed to as with units metes In tems of unit vectos = x xˆ + yyˆ If you ae at an initial position indicated by i and then late you ae at the final position denoted by You displacement vecto Whee = f - i f 2. Vectos in Physics 15
VELOCITY VECTORS (1) The aveage velocity vecto is defined as the displacement vecto divided by the elapsed time v = t units m/s, and is also a vecto ˆ av Since t is a scala, the aveage velocity vecto and displacement vecto ae paallel Example: A dagonfly is obseved initially at the position = (2.0m) xˆ + (3.5m) ŷ i. Thee seconds late it is at the position f = (-3.0m) xˆ + (5.5m) ŷ. What was the dagonfly s aveage velocity duing this time? 2. Vectos in Physics 16
VELOCITY VECTORS (2) Imagine a paticle moving in 2D along the blue path as shown If the paticle is at P 1 at time t 1 and at P 2 at time t 2, its displacement vecto is indicated by the vecto The aveage velocity is paallel to because on aveage the paticle has moved in the diection of duing the time fom t 1 to t 2 Fo smalle time intevals, the instantaneous velocity vecto is consideed v = lim t t 0 Example: Find the speed and diection of motion fo a paticle whose velocity is = (3.7m/s) + (-1.3m/s) v xˆ ŷ 2. Vectos in Physics 17
VELOCITY VECTORS (3) The instantaneous velocity vecto is obtained the aveage velocity vecto ove smalle and smalle time intevals In the limit of vanishing small time intevals, the aveage velocities appoach the instantaneous velocity Notice that the instantaneous velocity vecto points in the diection of motion at any given time 2. Vectos in Physics 18
ACCELERATION VECTORS (1) The aveage acceleation vecto ove an inteval of time is defined as the change in the velocity vecto divided by the time inteval a units m/s 2 av = v t The figue below shows the initial and final velocity vectos coesponding to two diffeent times Since v = v f v i and v f = v i + v The change in velocity vecto extends fom the head of the initial velocity vecto to the head of the final velocity vecto The diection of the aveage acceleation vecto is the same as that of the change in velocity vecto, since it is multiplied by 1/ t 2. Vectos in Physics 19
ACCELERATION VECTORS (2) An object can acceleate even if its speed is constant, povided its diection changes Suppose the initial velocity of the ca above is 12m/s and 10.0s late its final velocity is -12m/s ŷ The speed is 12m/s in each case, but the velocity is diffeent because its diection has changed Calculating the aveage acceleation gives a nonzeo acceleation a = /10.0s av = v t ( v f v) i Gives (-1.2m/s 2 ) + (-1.2m/s 2 ) xˆ A change in diection is just as impotant as a change in speed in poducing an acceleation Instantaneous acceleation a = lim v t (m/s 2 ) ŷ t 0 xˆ 2. Vectos in Physics 20
ACCELERATION VECTORS (3) The velocity vecto is always in the diection of a paticle s motion Howeve the acceleation vecto can point in diections othe than the diection of motion, and it geneally does Fo the figue above, at point 1 the paticle is slowing down At point 2, it is tuning left At point 3, it is tuning ight At point 4, it is speeding up 2. Vectos in Physics 21
RELATIVE MOTION Vectos can be used in descibing elative motion Suppose you ae at a platfom and a tain goes by at 15m/s On boad the tain a passenge is walking in the fowad diection at 1.2m/s elative to the tain How fast is the passenge moving elative to you? The answe is 15 + 1.2 = 16.2m/s What if the passenge had been walking at the same speed but towads the back of the tain? The new speed elative to you is -1.2 + 15 = 13.8m/s v tg If is the velocity of the tain elative to the gound v v pg pt is the velocity of the passenge elative to the tain is the velocity of the passenge elative to the gound Then v pg = v pt + v tg 2. Vectos in Physics 22
RELATIVE VELOCITY IN TWO DIMENSIONS Conside the situation when a peson climbs up a ladde on a moving tain with velocity v elative to pt the tain The tain moves elative to the gound with velocity Thus the velocity of the peson elative to the gound is the same as befoe v pg = v Example: Suppose the passenge on the tain shown above is climbing a vetical ladde with a speed of 0.2m/s, and the tain is coasting fowad at 0.7m/s. Find the speed and diection of the passenge elative to the gound. pt + v tg v tg 2. Vectos in Physics 23
NOTATION AND SUBSCRIPTS The appopiate subscipts ae used when efeing to elative motion Fo the plot above: v = 13 v12 v 23 Refeing to the figue above, the tail of each vecto is labelled with its fist subscipt, and the head of each vecto with its second subscipt Revesing the subscipts eveses the velocity as shown below + 2. Vectos in Physics 24
NOTATION AND SUBSCRIPTS: EXAMPLE You ae on a boat whose speed elative to the wate is 6.1m/s. The boat points at an angle of 25 upsteam on a ive flowing at 1.4m/s. What is you velocity elative to the gound? 2. Vectos in Physics 25