94. CHAPTER 3 Vectors and Motion in Two Dimensions. SUMMARY. The goals
of Chapter 3 have been to learn more about vectors and to use vectors as a tool ...
94
CHAPTER
3
Vectors and Motion in Two Dimensions
SUMMARY The goals of Chapter 3 have been to learn more about vectors and to use vectors as a tool to ana lyze motion in two dimensions.
GENERAL PRINCIPLES
Projectile Motion
Circular Motion
A projectile is an object that moves through the air under the innuence of grav ity and nothing else.
The motion consists of two pieces: I. Vert ical motion with free-fall accele rati o n.
0.,,= - g. 2. Horizon tal moti on with consta nt veloc ity.
The path of the motion is a parabola.
Kinemati c equations:
y
= + (vx)i Do l (\I.\.)f = (Vdj = constant )'f = )' j + (vy)j Dol - tg(Dor)2 (V,)r = (v,), - g t. r Xr
17 ~ , '-' (v..) j = v, cosO
- - - - - -"'--
For an object mov in g in a c ircle at a co nstam speed: The period T is the time for one rotation. The freq uency J= l IT is the number of revo lu tions per second.
Xj
The veloc ity is tangent to the c ircu lar path . The acce leration points toward the center of the circle and has magn itude
v'
a =-
r
IMPORTANT CONCEPTS
Vectors and Components A vector can be decomposed into x- and y-components.
The Acceleration Vector The sign of the components depends on the direction of the vector:
We define the acceleration vector as _ Vf- i\ Do li a = - - -=-
)'
Y Magnitude A
= VA
2
+A2
tf -
\' ;.
' A=ASintl o ~ I
_ _ _ _ _ _ _ _ .J
tj
Dot
We find the acceleration vector on a motion diagram as follows:
)'
Dots show positions at equal time interval:..
A,= Acos8
+---+--------x Di rection of A tI = lun- I(A I A)
Velocity vecto,""
go dot to dot.
" ..
The magnitude and direction of a vector can be expressed in terms of its components.
"r\C
_
~ I'r
~
II ~< O
The acceleration /':,.v vector points in the direction of l!.~
11,< 0
V"-ii"i
The difference in the veloci1Y vector. is found by addin g the negative orli, to vL•
APPLICATIONS Relative motion Velocities can be ex pressed re lative to an observer. We can add relat ive veloc ities to convert to another observer 's point of view.
c
= car, r =
runner, g = gro und
Car ~ _~1;;. 5 .;;n;;. '/';·;....._
tJ: ...... 5 m/'
Runner ,
The speed or the car with respect to the ru nner is:
Motion on a ramp An object sliding down a ramp will accelerate parall el to the ramp: {/~, =
± gsin8
The correct sign depends o n the direction in which the ramp is tilted.
y
",
~~x Same angle
Question s
r-~TM
IMP
For homework assigned on MasteringPhysics, go to . h· www.mastenngpyslcs.com
Problems la be led chapters;
BID
INT
95
integrate significant material from earlier
are of biological or medical interest.
Problem difficulty is labeled as I (straightforward) to 11111 (c halleng ing),
QUESTIONS Conceptual Questions I. a. Can a vecto r have non zero magnitude if a component is zero? If no, why not? If yes , give an exampl e. b. Can a vector have zero mag nitude and a nonzero compo-
nent? If no, why not? tr yes, give an example. 2. Is it poss ible to add a scalar to a vec tor? If so, demonstrate. If not, exp lain why nolo 3. Suppose two vectors have unequal magn itudes. Can their sum be O? Explain. 4. Suppose C= A + B a. Under what circumstances does C = A + B? b. Could C = A - B? Ir so, how? Ir not , why not? 5. For a projectile, which of th e foll ow in g quantiti es are constant durin g the ni ght: x, y, Vx' vy• v, a,," ay? Which o r the quantities are ze ro throughout the fli ght? 6. A baseball player throws a ball at a 40° angle to the ground. The ball lands on the ground some di stance away. a. Is there any point on the trajectory where vand a are parall el to each other? If so, where ? b. Is there an y point where v and (i are perpendi cular to eac h other? If so, where? 7. An athlete performin g the lo ng jump tri es to achi eve the maximum di stance from the point of takeoff to the first point of touchin g the ground . After the jump, rather than land upri ght , she ex tends her legs rorward as in the photo. How does thi s affect the time in the air? How does thi s give the jumper a longe r range? 8. A perso n trying to throw a ball as far as possible will run for· ward durin g the th.row. Explain why thi s increases the di stance of the throw. 9. A passe nger o n a jet airplane claims to be able to walk at a speed in excess of 500 mph . Can thi s be Hue? Exp lain. 10. If yo u go to a ski area, yo u' ll likely find th at the beginner's slope has the smallest angle. Use the concept of acceleration on a ramp to ex plain why this is so. II. In an amusement· park ride, cars rollin g alo ng at hi gh speed suddenl y head up a lo ng, strai ght ramp. They roll up the ramp, reverse direction at the hi ghe st poi nt, the n roll backward bac k down the ramp. In eac h o r the following segments o r the motion, are the cars accelerating, or is their acceleration zero? If accelerating, which way does their acceleration vector point ? a. As the cars roll up the ramp. b. At the hi ghest point on the ramp. c. As the cars roll back down the ramp.
12. There are competition s in which pilots n y small planes low over the ground and drop weights, trying to hit a target. A pilot nyin g low and slow drops a weight; it takes 2.0 s to hit the ground , duri ng which it travel s a hori zontal di stance of 100m . Now the pilot does a run at the same height but twice the speed. How much time does it take the we ight to hit the ground? How far does it travel before it lands ? 13. A cyclist goes around a leve l, circul ar track at constant speed. Do you agree or di sagree with the following statement: "Because the cyclist's speed is constant, her acceleration is zero. " Explain. 14. You are driving yo ur car in a circular path on level ground at a constant speed o r 20 mph. At the in stant you are driving north , and turning left, are you acce leratin g? .If so, toward what point of the co mpass (N, S, E, W) does yo ur acce leration vec to r point? !f not, why not? 15 . An airplane has been directed to fly in a clockwi se circle, as see n from above, at constant speed until another plane has landed. Whe n the pl ane is goin g north, is it acce leratin g? If so, in what direction does the acceleration vector point ? If not, why not? 16. Whe n yo u go around a corner in your car, your car foll ows a path that is a seg ment of a circle. To tum safel y, you should keep your car 's acce lerat ion below some safe upper limit. If you want to make a " ti ghter" turn- that is. turn in a c ircle with a smaller radius-how should you adjust your speed? Explain.
Multiple-Choice Questions 17. II Which combinatio n of the vec tors shown in Fig ure Q3. 1.7 has the largest magn itude ?
A+B+C SB +A-C C. A-B+C D. C- A- C A.
Ii
FIGURE 0 3.17
18. II Two vectors appear as in Fig ure Q3. 18. Whi ch combi nati on points directly to the left?
P+Q P- Q c. Q-P D. -Q- P
A.
8.
y
~
FIGURE 0 3.18
19. I The gas pedal in a car is somet imes refelTed to as ;;the accel· erator." Which other controls on the vehicle can be used to produce accelerati on? A. The brakes. B. The steerin g wheel. C. The gear shift. D. All of the above.
96
CHAPTER 3
Vectors and Motion in Two Dimensions
20. I A car travel s at cons tant speed along the curved path shown from above in Figure Q3.20. Five poss ible vectors are also shown in the figure; the letter E represents the zero vector. Which vector best represen ts a. The car's velocity at position I? b. The car's acceleration at point I? c. The car's velocity at position 2? d. The car's acceleration at point 2? e. The car's velocity at position 3? r. The car 's acceleration at point 3?
FIGURE 03 .22
23. I A cannon. ele vated at 40° is fired at a wall 300 m away on level ground, as shown in Fig ure Q3.23. The initial speed of the cannon ball is 89 m/s
2
Vi
= 89 m}s
40' 300m
FIGURE 03 .23 3
FtGURE 03.20
21. I A baU is fired from a cannon at point I and follows the trajectory shown in Figure Q3.21. Air resistance may be neglected. Fi ve possible vectors are also shown in the fi g ure; the letter E represents the zero vector. Which vector best represents a. The ball 's velocity at position 2? b. The ball 's acceleration at point 2? c . The ball's velocity at position 3? d. The baU 's acceleration at point 3?
24.
3
25.
2
A
26.
FtGURE 03 .21
22. I A ball thrown at an initial angle of 37.0° and initial veloc ity of 23.0 mls reaches a maximum height II, as shown in Fi gure Q3.22. With what initial speed musl a ball be thrown straight lip to reach the same maximum height It? A. 13.8 m/s B. 17.3 m/s c. 18.4 m/s D. 23.0 m/s
27.
a. How long does it take for the ball to hit the wall? A. 1.3 s B. 3.3 s C. 4.4 s D. 6.8 s E. 7.2 s b. At what he ight II does the ball hit the wall? A. 39 m B. 47 m C. 74 m D. 160m E.21Om A car drive s horizontally off a 73- m-hi gh cliff at a speed of 27 m/s. Ignore air resistance. a. How long will it take the car to hit the ground? A. 2.0 s 8. 3.2 s C. 3.9 s D. 4.9 s E. 5.0s b. How far from the base of the clifrwi ll lhe car hit ? A.74m 8.88 m C. 100m D. 170m E. 280m I A football is kicked at an angle of 30° with a speed of20 m/s. To the nearest seco nd, how long will the ball stay in the air? A. I s B. 2 s C. 3 s D. 4 s I A football is kicked at an angle of 30° with a speed of 20 m/s. To the nearest 5 m, how far wili lhe ball travel? A. 15 m 8. 25 m C. 35 In D. 45 m I Riders o n a Ferris wheel move in a c ircle with a speed of 4.0 m/s. As they go around, they experience a centripetal acceleration of 2.0 m/s 2 • What is the diameter of thi s particular Ferris wheel? A.4.0m B.6.0 m C.8.0m D. 16m E.24m
PROBLEMS Section 3.1 Using Vectors
Section 3.2 Using Vectors on Motion Diagrams
1. II Trace the vectors in Figure P3. 1 onto your paper. Then usc graphical methods to draw the vectors (a) Ii + 8 and (b) A - 8. FIGURE P3 .1
2. III Trace Ihe vecto rs in Fi gure P3 .2 onto your paper. Then li se graphica l methods to draw the vectors (a) Ii + Ii and (b) A - 8. FIGURE P3 .2
3. I A car goes around a corner in a circular arc at constant speed . Draw a motion diagram includin g positi o ns, velocity vectors, and acceleration vectors. 20 4. I a. Is (he object's average speed between points I and 2 greater than, less than, or equal to it s average speed between points 0 \ 0 and I? Explain how you can tell. b. Find the average acceleration vec tor at 0 point 1 of the three-point mot ion dia- 0 gram in Figure P3.4. FIGURE P3 .4
Problems 5. 1111 Figure 3. 11 showed the motion diag ram for Anne as she rode a Ferri s wheel that was turn in g at a constant speed. The in set to the figure showed how to rind the acce le ration vee· tor at the lowest point in her motion. Use a similar analys is to find Anne ' s accelerati on vector at the 12 o' clock, 4 o'clock, and 8 o ' clock po siti ons of the motion diagram. Use a rul e r so that yo ur ana lysis is accurate.
Section 3.3 Coo rdinate Systems and Vector Components 6. II A position vector with magn itude to m points to the ri ght and
up. Its x-component is 6.0 m. What is !.he val ue of its y-component ? 7. III A ve locity vector 40" above the pos itive x-axis ha s a ycom ponen t of 10 m/s. What is the val ue of its x-componen t? 8. II Jack and Jill ran up the hill at 3.0 m/s. The horizontal component of J ill 's velocity vec tor was 2.5 m/s. a. What was the angle of the hill ? b. What was the vertical component of Jill's velocity? 9. II A cannon tilted upward at 30" fires a ca nnonball with a speed of 100 m/s. At th at in stant , what is the compo nen t of the cannonball' s veloc ity parall el to the ground ? y 10. 11 a. What are the x- and y-components of vector if. of Figu re P3.10 in terms of the angle (J and the magnitude £? b. For the same vector, what are the x- and y-componen ts in tenns of the angle
53. 1111 A skier is gliding along at 3.0 mls on horizontal, frictionless snow. He suddenly starts dow n a 10° incline. Hi s speed at the bottom is 15 Ill/s. a. What is the length of the incl ine? b. How long docs it take him to reach the bollom ? 54. III A block slides a long the frictionless track shown in 45° 30 Figure P3.54 with an ini tial 1--------01 - - - - - - speed o f 5.0 m/s. Ass ume it I.Om turns aU the corners smoothly, FIGURE P3 .54 with no loss of speed. a. What is the block· s speed as it goes over the top? b. What is its speed when it reaches the leve l track on the right side? c. By what perce ntage does the block' s final speed differ from its initial speed? Is thi s surpri sing? 55. III One game at the amuse,m ent park has you push a puck up a long, frictionless ramp. Yo u win a stuffed animal if the puck , at its hi ghest point, comes to within 10 cm o f the end of the ramp without going off. You g ive the puck a push, releasing it with a speed of 5.0 m/s when it is 8.5 m from the end of the ramp_The puck 's speed after trave ling 3.0 m is 4.0 m1s. Are you a winner? 56. 1111 When the moving sidewalk at the airport is broken, as it often seems to be, it takes YOll 50 s to wa lk from your gate to the baggage claim. When it is working and YOll stand on the mov ing sidewalk the entire way, without wa lking, it takes 75 s to travel (he same di stance. How long will it take YO ll to travel from the gate to baggage claim if you walk while riding on the moving sidewalk ? 57. 11111 Ships A and B leave pon together. Fo r the nex t two hours, ship A travels al 20 mph in a direc ti o n 30° west of north while ship B trave ls 20° cast o f nort h at 25 mph . a. What is the di stance betwee n the two s hips two hours after they depart ? b. What is the speed of ship A as seen by ship B? 58. III Mary needs to row her boat across a l OO-m-wide river that is flowing to the east at a speed of 1.0 m/s . Mmy can row the boat with a speed of2.0 m/s relati ve to the water. a. If Mary rows straight north, how far downstream will she land? b. Draw a picture showing Mary's d is place ment d ue to rowing. her di splacement due to the river's motio n, and her net d isplacement. 59. III A nock of ducks is try ing to migrate south for the winte r, but they keep bein g blow n off course by a wind blowing from the west at 12 m/s. A wise elder duck finally real izes that the solution is to fly at an angl e to the wind. If the ducks ca n fly at 16 mls relat ive (0 the air, in what directio n shou ld they head in order to move directly south ?
~ .o m
g
100
CHAPTER 3
Ve ctors and Moti o n in Two Dimen s io ns
60. III A kayaker needs to paddle north across a 1OO- m-w ide harbor. T he tide is go ing ou t, creat ing a tidal current now in g east at 2.0 m/s. Th e kayaker can paddle with a speed o f 3.0 mls. a. In w hi ch d irecti on s hould he padd le in orde r to travel straight across the harbor? b. How long will it take him to cross? 6 1. 1111 A plane has an airspeed of 200 mph. The pi lot wishes to reach a destin at io n 600 mi d ue east, b ut a win d is blowi ng at 50 mph in the d irec tion 30° north o f east. a. In what d irect ion must the pil ot head the plane in order to reach her desti nati on? b. How long will the tri p take? 62. III The G ulf S tream o ff the eas t coast o f the Un ited States can flow at a rap id 3.6 mls to the north. A ship in thi s c urre nt has a crui s ing speed o f 10 m/s. T he ca pta in wo ul d li ke to reac h la nd at a po int due west fro m the current pos itio n. a. In what directio n with respect to the water should the ship sail ? b. At this head ing, what is the ship's speed with respect to land? 63. II A phys ics stude nt on Pl anet y Ex idor throws a ball , and it foll .Os lows the parabolic trajectory 1.0 s 3.0s shown in Figure P3.63. The ball 's position is shown at 1.0 s in terva ls un til { = 3.0 s. At { = 1.0 s, the baU's veloc ity has compo- ol
