Physics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving

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1 Physics 11 Chapte 3: Vectos and Motion in Two Dimensions The only thing in life that is achieved without effot is failue. Souce unknown "We ae what we epeatedly do. Excellence, theefoe, is not an act, but a habit. Aistotle Act as if what you do makes a diffeence, because it does. Souce unknown Reading: pages 67 95; skip sections 3.5 and 3.8 Outline: using vectos (coveed on PowePoint slides) vecto addition multiplication by a scala vecto subtaction using vectos in motion diagams (ead on you own) coodinate systems and vecto components adding vectos using components motion on a amp motion in two dimensions pojectile motion solving poblems lots of examples Poblem Solving Vectos can be given eithe in tems of a magnitude and diection o in tems of components; answes may be equested in eithe of these foms. This means you may need to convet fom one fom to anothe. If you know the components of a vecto, you can easily find the magnitude and diection. Fo a vecto a with components ax and ay, the magnitude is given by a = a + a 2 2 x y and the angle is given by ay θ = actan ax Afte you have found θ, check to be sue that θ is in the coect quadant. If it is not, then add 180 to the value of the angle you get fom the calculato. Pojectile motion poblems ae the same as poblems fom chapte 2 except that you have two sets of equations, one fo the hoizontal motion and one fo the vetical motion. Remembe that fo pojectile motion, ax = 0 and ay = -9.8 m/s 2.

2 The equations that descibe the hoizontal motion ae: xf = xi + (vx)i t (vx)f = (vx)i = constant The equations that descibe the vetical motion ae: y f = yi + (vy)i t + ½ ay( t) 2 (vy)f = (vy)i + ay( t) (v y )f 2 = (vy)i 2 + 2ay y If up is defined as positive, then a y = -g = -9.8 m/s 2. If down is defined as positive, then ay = 9.8 m/s 2. If the initial speed and launch angle of the pojectile ae given, then (vx)i = vi cos(θ) and (v y )i = vi sin(θ), assuming θ is given with espect to the +x axis. One of the key things to ealize when dealing with two dimensional motion is that you can teat each dimension sepaately. The x pat of the motion and the y pat of the motion ae completely independent of each othe; each behaves as if the othe did not exist. Howeve, the two motions ae connected by the vaiable t. Mathematical Skills Tigonomety is a lage potion of the mathematics used in this chapte. Hee is a listing of the impotant elements you should know vey well. The Pythagoean theoem The squae of the hypotenuse of a ight tiangle equals the sum of squaes of the othe two sides. In the diagam C 2 = A 2 + B 2. The theoem is tue only if the tiangle contains a ight angle (90 ). The theoem is used, fo example, to calculate the magnitude of a vecto in the xy plane given its x and y components. Tigonometic functions. Fo the tiangle shown, A = C cosθ and B = C sinθ. The elations follow diectly fom the definition of the sine and cosine and ae used to find the components of a vecto, given the magnitude and the angle it makes with an axis. Also know that fo the tiangle above tanθ = B/A. This elationship, in the fom θ = actan(ay/ax), is used to find the angle a vecto makes with a coodinate axis. WARNING! Fo any values of ax and ay the equation θ = actan(ay/ax) has two solutions fo θ. If you use a calculato to evaluate θ, it will only give you an angle in the 1 st o 4 th quadant. If you know that the angle must be in the 2 nd o 3 d quadant (because of the signs of the x- and y- coodinates), then you must add 180 o to the angle given by the calculato.

3 SUMMARY The goals of Chapte 3 have been to lean moe about vectos and to use vectos as a tool to analyze motion in two dimensions. GENERAL PRINCIPLES Pojectile Motion A pojectile is an object that moves though the ai unde the influence of gavity and nothing else. The path of the motion is a paabola. (v y ) i 5 v i sin u y u (v x ) i 5 v i cos u v i The motion consists of two pieces: 1. Vetical motion with fee-fall acceleation, a y =-g. 2. Hoizontal motion with constant velocity. Kinematic equations: x x f = x i + (v x ) i t (v x ) f = (v x ) i = constant y f = y i + (v y ) i t g( t)2 (v y ) f = (v y ) i - g t Cicula Motion Fo an object moving in a a cicle at a constant speed: The peiod T is the time fo a one otation. The fequency f = 1/T is the numbe of evolutions pe v second. The velocity is tangent to the cicula path. The acceleation points towad the cente of the cicle and has magnitude a = v2 v a Centipetal acceleation v IMPORTANT CONCEPTS Vectos and Components A vecto can be decomposed into x- and y-components. y Magnitude A 5 "Ax 2 1 A y 2 u A x 5 A cos u Diection of A u 5 tan 21 (A y / A x ) A Ay 5 A sin u The magnitude and diection of a vecto can be expessed in tems of its components. x The sign of the components depends on the diection of the vecto: A x, 0 A x, 0 A y. 0 A A A y, 0 y A y. 0 A A y, 0 A A x. 0 A x. 0 x The Acceleation Vecto We define the acceleation vecto as a B = vb f - v B i t f - t i = vb t We find the acceleation vecto on a motion diagam as follows: Dots show positions at equal time intevals. Velocity vectos go dot to dot. v i a v f The acceleation vecto points in the diection of Δv. Δv v f 2v i The diffeence in the velocity vectos is found by adding the negative of v i to v f. APPLICATIONS Relative motion Velocities can be expessed elative to an obseve. We can add elative velocities to convet to anothe obseve s point of view. c = ca, = unne, g = gound Ca Runne 5 m/s 15 m/s The speed of the ca with espect to the unne is: 1v x 2 c = 1v x 2 cg + 1v x 2 g Motion on a amp An object sliding down a amp will acceleate paallel to the amp: a x = g sin u The coect sign depends on the diection in which the amp is tilted. y a fee fall u a x a y u Same angle x

4 Question 1 Questions and Example Poblems fom Chapte 3 (a) Can a vecto have nonzeo magnitude if a component is zeo? If no, why not? If yes, give an example. (b) Can a vecto have zeo magnitude and a nonzeo component? If no, why not? If yes, give an example. Question 2 Vecto A points along the +y axis and has a magnitude of units. Vecto B points at an angle of 60.0 o above the +x axis and has a magnitude of units. Vecto C points along the +x axis and has a magnitude of units. Which vecto has (a) the lagest x component and (b) the lagest y component? Question 3 Fo a pojectile, which of the following quantities ae constant duing the flight: x, y, v x, vy, v, ax, ay? Which o the quantities ae zeo thoughout the flight? Question 4 A tennis ball is hit upwad into the ai and moves along an ac. Neglecting ai esistance, whee along the ac is the speed of the ball (a) a minimum and (b) a maximum? Justify you answes. Question 5 The figue shows thee paths fo a football kicked fom gound level. Ignoing the effects of ai esistance, ank the paths accoding to (a) time of flight, (b) initial vetical velocity component, (c) initial hoizontal velocity component, and (d) initial speed.

5 Poblem 1 Tace the vectos in the figue below onto you pape. Then use gaphical methods to daw the vectos (a) A+ B and (b) A B. Poblem 2 Daw each of the following vectos, then find its x- and y-components. a) d = (2.0 km, 30 left of +y-axis) b) v = (5.0 cm/s, - x-diection) c) a = (10 m/s 2, 40 left of -y-axis)

6 Poblem 3 While visiting England, you decide to take a jog and find youself in the neighbohood shown on the map in the figue below. What is you displacement afte unning 2.0 km on Stawbey Fields, 1.0 km on Penny Lane, and 4.0 km on Abbey Road? Poblem 4 A golfe, putting on a geen, equies thee stokes to hole the ball. Duing the fist put, the ball olls 5.0 m due east. Fo the second out, the ball tavels 2.1 m at an angle of 20.0 o noth of east. The thid put is 0.50 m due noth. What displacement (magnitude and diection elative to east) would have been needed to hole the ball on the vey fist put?

7 Poblem 5 A piano has been pushed to the top of the amp at the back of a moving van. The wokes think it is safe, but as they walk away, it begins to oll down the amp. If the back of the tuck is 1.0 m above the gound and the amp is inclined at 20, how much time do the wokes have to get to the piano befoe it eaches the bottom of the amp? Poblem 6 A dive uns hoizontally with a speed of 1.20 m/s off a platfom that is 10.0 m above the wate. What is the speed just befoe stiking the wate?

8 Poblem 7 The punte on a football team ties to kick a football so that it stays in the ai fo a long hang time. (a) If the ball is kicked with an initial velocity of 25.0 m/s at an angle of 60.0 o above the gound, what is the hang time? (b) How fa does the ball tavel befoe it hits the gound? Poblem 8 A golf ball is stuck at gound level. The speed of the golf ball as a function of the time is shown in the figue below, whee t = 0 at the instant the ball is stuck. (a) How fa does the golf ball tavel hoizontally befoe etuning to gound level? (b) What is the maximum height above gound level attained by the ball?

9 Poblem 9 A ca dives staight off the edge of a cliff that is 54.0 m high. The police at the scene of the accident note that the point of impact is m fom the base of the cliff. How fast was the ca taveling when it went ove the cliff? Poblem 10 A golfe, standing on a faiway, hits a shot to a geen that is elevated 5.50 m above the point whee she is standing. If the ball leaves he club with a velocity of 46.0 m/s at an angle of 35.0 o above the gound, find the time the ball is in the ai befoe it hits the geen.

10 Poblem 11 King Athu's knights use a catapult to launch a ock fom thei vantage point on top of the castle wall, l2.0 m above the moat. The ock is launched at a speed of 25 m/s and an angle of 30.0 above the hoizontal. How fa fom the castle wall does the launched ock hit the gound?