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 The motion consists of two pieces:

Circular Motion A projectile is an object that moves through the air under the innuence of grav ity and nothing else.

I. Vert ical motion with free-fall accele ration. 0.,,= - g.

For an object moving in a c ircle at a constam speed:

The period T is the time for one rotation.

The path of the motion is a parabola.

2. Horizon tal motion with constant veloc ity.

Kinemati c equations: The freq uency J= l IT is the number of revo lu tions per second. y

Xr = Xj + (vx)i Do l The veloc ity is tangent to the c ircu lar path .

(\I.\.)f = (Vdj = constant 17 ~ , )'f = )' j + (vy)j Dol - tg(Dor)2

The acce leration points toward the center of the circle and has magn itude

'-' (v .. ) j = v, cosO ------"'-- (V,)r = (v,), - g t. r

IMPORTANT CONCEPTS

Vectors and Components

v' a = ­

r

The Acceleration Vector A vector can be decomposed into x- and y-components.

The sign of the components depends on the direction of the vector:

We define the acceleration vector as

_ Vf- i\ Do li

Y Magnitude A = VA 2 + A 2

\' ;. ~' A=ASintl o I )'

________ .J

)'

A,= Acos8

+---+--------x Di rection of A tI = lun- I(A I A) " ..

The magnitude and direction of a vector can be expressed in terms of its components. II~< O 11,< 0

APPLICATIONS

Relative motion Velocities can be expressed 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 = ground

Car ~ _~1 ;;.5 .;;n;;.'/';;.· .... _

tJ: 5 m/' Runner , ......

The speed or the car with respect to the ru nner is:

a = - - -=-t f - t j Dot

We find the acceleration vector on a motion diagram as follows:

Dots show positions at equal time interval:..

Velocity vecto,"" go dot to dot.

Motion on a ramp An object sliding down a ramp will accelerate parall el to the ramp:

{/~, = ± gsin8

The correct sign depends on the direction in which the ramp is tilted.

"r\C _ ~ I'r

The acceleration /':,.v ~ vector points in V"-ii"i the direction of l!.~

y

The difference in the veloci1Y vector. is found by adding

the negative orli, to vL•

",

~~x Same angle

r--~TM For homework assigned on MasteringPhysics, go to IMP . h· www.mastenngpyslcs.com

Problem difficulty is labeled as I (straightforward) to 11111 (challenging),

QUESTIONS

Conceptual Questions

I. a. Can a vecto r have nonzero magnitude if a component is zero? If no, why not? If yes , give an example.

b. Can a vector have zero magnitude 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, explain 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 the following quantiti es are constant during the ni ght: x, y, Vx' vy• v, a,," ay? Which or 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 v and a are parallel

to each other? If so, where? b. Is there an y point where v and (i are perpendicular to each

other? If so, where? 7. An athlete performing the long

jump tri es to achieve the maxi­mum distance from the point of takeoff to the first point of touch­ing the ground. After the jump, rather than land upright , 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 longer range?

8. A person trying to throw a ball as far as possible will run for· ward during the th.row. Explain why thi s increases the di stance of the throw.

9. A passenger on a jet airplane claims to be able to walk at a speed in excess of 500 mph. Can thi s be Hue? Explain.

10. If yo u go to a ski area, you' ll likely find that the beginner's slope has the smallest angle. Use the concept of acceleration on a ramp to explain why this is so.

II. In an amusement· park ride, cars rolling along at high speed suddenly head up a long, straight ramp. They roll up the ramp, reverse direction at the highest point, then roll backward back down the ramp. In each or the following segments or 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 highest point on the ramp. c. As the cars roll back down the ramp.

Questions 95

Problems labe led INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

12. There are competitions in which pilots n y small planes low over the ground and drop weights, trying to hit a target. A pilot nying low and slow drops a weight; it takes 2.0 s to hit the ground, duri ng which it travels 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, circular 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 your car in a circular path on level ground at a constant speed or 20 mph. At the instant you are driving north , and turning left, are you acce lerating? .If so, toward what point of the compass (N, S, E, W) does your acce leration vec tor point? !f not, why not?

15 . An airplane has been directed to fly in a clockwise circle, as seen from above, at constant speed until another plane has landed. When the plane is going north, is it accelerating? If so, in what direction does the acceleration vector point? If not, why not?

16. When you go around a corner in your car, your car follows a path that is a segment 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 combination of the vec tors shown in Figure Q3. 1.7 has the largest magnitude?

A. A +B+C SB +A-C C. A-B+C D. C- A- C

Ii

FIGURE 0 3.17

18. II Two vectors appear as in Figure Q3. 18. Which combi nation points directly to the left?

A. P+Q y ~ 8. P- Q c. Q-P D. -Q- P 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 acceleration? A. The brakes. B. The steering wheel. C. The gear shift. D. All of the above.

96 CHAPTER 3 Vectors and Motion in Two Dimensions

20. I A car travels 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?

2

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 gure; 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?

3

2

A

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 Figure 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

PROBLEMS

Section 3.1 Using Vectors

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.

2. III Trace Ihe vecto rs in Figure P3 .2 onto your paper. Then li se graphi­ca l methods to draw the vectors (a) Ii + Ii and (b) A - 8.

FIGURE P3 .1

FIGURE P3 .2

FIGURE 03 .22

23. I A cannon. elevated at 40° is fired at a wall 300 m away on level ground, as shown in Figure Q3.23. The initial speed of the cannonball is 89 m/s

24.

Vi = 89 m}s

40'

300m

FIGURE 03 .23

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 drives 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

25. I A football is kicked at an angle of 30° with a speed of20 m/s. To the nearest second, how long will the ball stay in the air? A. I s B. 2 s C. 3 s D. 4 s

26. I A football is kicked at an angle of 30° with a speed of 20 m/s.

27.

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 on 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 D. 16m

B.6.0 m E.24m

C.8.0m

Section 3.2 Using Vectors on Motion Diagrams

3. I A car goes around a corner in a circular arc at constant speed . Draw a motion diagram including positions, velocity vectors, and acceleration vectors.

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 and I? Explain how you can tell.

b. Find the average acceleration vec tor at point 1 of the three-point mot ion dia­gram in Figure P3.4.

20

\ 0

0 0

FIGURE P3 .4

5. 1111 Figure 3. 11 showed the motion diag ram for Anne as she rode a Ferri s wheel that was turn ing at a constant speed. The inset to the figure showed how to rind the acce leration 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 positions of the motion diagram. Use a ruler so that your ana lysis is accurate.

Section 3.3 Coordinate Systems and Vector Components

6. II A position vector with magnitude to m points to the right and up. Its x-component is 6.0 m. What is !.he value of its y-component?

7. III A ve locity vector 40" above the pos itive x-axis has a y­component of 10 m/s. What is the val ue of its x-component?

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 cannonball with a speed of 100 m/s. At that instant , what is the component of the can­nonball' s veloc ity parallel to the ground?

10. 11 a. What are the x- and y-components of vec­tor if. of Figure P3.10 in terms of the angle (J and the magnitude £?

b. For the same vector, what are the x- and y-components in tenns of the angle <p and the magnitude £?

y

~ , 11. Draw each of the following vectors, then find FIGURE P3 .l0

its x- and .v-components. a. d = (100 m, 45" below +x-axis) b. ii = (300 mIs, 20" above +x-axis) c. a = (5.0 I11/S2, -y-direct ion)

12. II Draw each o f the following vectors, then find its x- and .v-components. a. d = (2.0 km, 30° left of +y-ax is) b. v = (5 .0 cntls, - x-direct ion) c. a = (10 m/s2, 40" le ft of - .v-axis)

13. I Each of the following vectors is given in terms of its x- and .v-components. Draw the vector, label an angle lhat specifies the vector's direction, then find the vector's magnitude and direction. a. Vt = 20 mis, \/,. = 40 m/s b. a,. = 2.0 m/s2,-a\" = -6.0 m/s2

14. I Each of the following vectors is given in terms of its x- and y-components. Draw the vector, label an angle that specifies the vector· s directi on, then find the vector 's magni tude and direction. a. v, = 10 mIs, V,. = 30 m/s

, . nI ' b. a.,. = 20 ntls-.Gy = 10 1 .. s-IS. 1111 While visiting England, you decide to take a jog and find

yourself in the neighborhood shown on the map in Figure P3.15. What is your di splacement after running 2.0 km on Strawberry Fields, 1.0 kIll on Penny Lane, and 4 .0 km on Abbey Road?

S",n I~

. ""-"7° FIGURE P3 .1S

End

•

Problems 97

Section 3.4 Motion on a Ramp

16. III You begin sliding down a IS" ski slope. Ignoring friction and air resistance, how fast will you be moving after lOs?

17. 1111 A car traveling at 30 m/s runs alii of gas wh ile travel ing up a 5.0" slope. How far will it coast before starting to roll back down?

18. II In the Soapbox Derby, young participants bu ild non­motori zed cars with very low­friction wheels. Cars race by rolling down a hill. The track at Akron's Derby Downs, where the national championship is held, beg ins with a 55- ft ~long

section tilted 13" below hori­zon tal. a. What is the maximum poss ible acceleration of a car moving

down thi s stretch of track? b. If a car starts from rest and undergoes thi s acceleration for

the full 55 ft, what is its final speed in mls? 19. 111 A piano has been pushed to the top of the ramp at the back of

a moving van . The workers think it is safe, but as they walk away, it begins to roll down the ramp. If the back of the truck is J.O m above the ground and the ramp is inclined at 20", how much time do the workers have to get to the piano before it reaches the bottom of [he ramp?

20. II Starting from rest, several toy cars roll down ramps of differ~ ing lengths and angles. Rank [hem accord ing to their speed at the bottom of the ramp, from slowest to fastest. Car A goes down a JO m ramp inclined at IS", car B goes down a 10 m ramp inclined at 20", car C goes down an 8.0 m ramp inclined at 20", and car D goes down a 12 m ramp inclined at 12".

Section 3,5 Relative Motion

21. I Anita is running to the right at 5 mIs, as shown in Figure P3.2 1. Balls I and 2 are thrown toward her at 10 m/s by friends standing on the ground. According to Anita, what is the speed of each balI?

22.

-' 5m/s

FIGURE P3 .2l

I Anita is running to the ri ght at 5 mIs, as shown in Figure P3.22. Balls I and 2 are thrown toward her by friends standing on the ground. According to Anita, both balls are approaching her at 10 m/s. Accord ing to her friends, with what speeds were the baHs thrown?

-" 5m/5

FIGURE P3 .22

98 CHAPTER 3 Vectors and Motion in Tw o Dimen sions

23. 1111 A boat takes 3.0 h to travel 30 km down a ri ver, then 5.0 h to return. How fas t is the river flow ing?

24. II Two chi ldre n who are bored while wa iting for the ir fli ght at the airport dec ide to race from one end of the 20-m-long mov­ing sidewalk to the other and back. Phillippe runs on the side­walk at 2.0 mls (re lati ve to the sidewalk) . Renee runs on the fl oor at 2.0 m/s. The sidewalk moves at 1.5 mls relati ve to the fl oor. Both make the turn instan Lly with no loss of speed. a. Who wins the race? b. By how much time does the wi nner win?

25. II A skydiver deploys his parachute when he is 1000 m direc tly above hi s del) ired landing spot. He then fa ll s through the air at a steady 5.0 m/s. There is a breeze blowing to the west at 2.0 m/s . a. At what angle with respec t to vertical does he fa ll? b. By what distance will he miss hi s des ired landing spot?

Section 3.6 Molion in 1\\'0 Dimensions: Projectile Motion

Section 3.7 Projectile Motion: Solving Problems

26. 1111 An object is launched with an initial veloc ity of 50.0 mls at a launch ang le of 36.9° above the horizontal. a. Make a table showing values of x, y , "x. v ... ' and the speed v

every 1 s from t = 0 s to t = 6 s. b. Plot a graph of the object 'S trajectory during the fi rst 6 s of

motion. 27. III A ball is thrown hori zontally from a 20-m-high building wi th

a speed of 5.0 m/s. a. Make a sketch of the ball 's trajectory. b. Draw a graph of 11.[. the hor izontaJ veloc ity, as a func tion of

time. Include un its on both axes. c. Draw a graph of v.'" . the vert ical veloci ty, as a func tion of

time. Include uni ts on both axes. d. How far from the base of the bui ld ing does the ba ll hit the

ground? 28. A ba ll with a hori zontal speed of 1.25 mls roUs off a bench

1.00 m above the fl oor. a. How long will it take the ball to hi t the floor? b. How far from a point on the floor direct ly be low the edge of

the bench wi ll the ba ll land? 29. Ilill Ki ng Arthur's kn ights use a catapult to launch a rock from

thei r vantage point on top of the castl e wall , l2 m above the moat. The rock is launched at a speed of 25 mls and an angle of 30° above the horizontal. How far from the cast le wall does the launched rock hit the ground?

30. I Two spheres are launched horizontally from a I.O- m-high table . Sphere A is launched with an ini tial speed of 5.0 m/s. Sphere B is lau nched with an ini tial speed of 2.5 mls. a. What are the times fo r each sphere to hi t the floor? b. What are the di stances that each trave ls from the edge of the

table? 3 1. III A rifle is aimed horizontally at a target 50 m away. The bullet

hi ts the target 2.0 cm below the aim point. a. What was the bullet's fl ight time? b. What was the bullet's speed as it left the barre l?

32. III A gray kangaroo can bound across a fl at stretch of ground 810 with each j ump can-ying it 10 m from the takeoff point. If the

kangaroo leaves the ground at a 20° angle, what are its (a) take­off speed and (b) horizo ntal speed?

33. III On the ApolJo 14 miss ion to the moon, astronaut Alan Shep­ard hi t a go lf ball with a go lf cl ub improvised from a tool. The

free-fall acce le ration on the moon is 1/6 of its va lue on earth . Suppose he hi t the ball with a speed of 25 mls at an angle 30° above the horizontal. a. How long was the ball in fl ight? b. How far did it travel? c. Ignoring air res istance, how much farther would it travel on

the moon than on earth?

Section 3.8 Motion in Two Dimensions: Circular Motion

34. An old~fashioned LP record rotates at33~ rpm. a. What is its frequency, in rev/s? b. What is its period, in seconds?

35. I A typical hard disk in a computer spins at 5400 rpm. a. What is the frequency, in rev/s? b. What is the period, in seconds?

36. I Rac ing greyhounds are capable of ro und ing corners at very 810 high speeds. A typical greyhound track has turns that are 45-m­

diameter semicircles. A greyhound can ru n around these turns at a constant speed of 15 m/s . What is its acceleration in m/s 2 and in units of g?

37. II A CD-ROM dri ve in a compu ter spins the 12-cm-diameter disks at 10,000 rpm. a. What are the di sk's period (in s) and frequency (in rev/s)? b. What would be the speed of a speck of dust on the ou tside

edge of thi s disk? c. What is the accelerat ion in uni ts of g that thi s speck of dust

experiences? 38. To withstand "g-forces" of up to 10 g's, caused by suddenly 810 pulli ng out of a steep dive, fighter jet pi lots train on a "human

cen tri fuge." 10 g's is an acceleration of 98 m/s2. lf the length of the cen trifuge arm is 12 m, at what speed is the ri der moving when she experiences 10 g's?

39. II A part icle rotates in a c ircle with centr ipetal acce le ration (/ = 8.0 m/s 2. What is (/ if a. The radi us is do ubled withou t chang ing the part icle's

speed? b. The speed is doubled without chang ing the c ircle's radi us?

40. II En tra nce and ex it ramps for freeways are often c ircular stretches of road. As you go around one at a constant speed, you will experience a constan t acceleration. Suppose YOll drive through an entrance ramp at a modest speed and YOllf acceleration is 3.0 m/s2. What will be the accelerati on if you double YOUi' speed?

4 1. II A pereg ri ne falcon in a tight, circular turn can atta in a cen­BID lripetal acce lerat ion 1.5 times the free- faU acceleration. If the

falcon is fly ing at 20 mis, what is the rad ius of the turn?

General Problems

42. I Suppose C = A + 8 wbere vector A has components A x = 5, Ay = 2 and vector 8 has components Bx = -3, By = -5.

a. What are the x- and y-components of vector C? b. Draw a coord inate system and on it show vectors A, 8,

and C. c. What arc the mag ni tude and direction of vector C?

43.

44.

45.

Suppose jj = A - jj where vector A has components Ax = 5. Ay = 2 and vector Ii bas components Bx = -3. By = -5.

a. What are the x- and y-components of vector O? b. Draw a coordinate system and on it show vectors A, 8, and 0, c. What arc the magn itude and direction of vector en II Suppose E = 2A + 38 where vector A has components A.f = 5,

AI' = 2 and vector jj has components fix = -3, BI • = - 5.

a. What arc the x- and y-componcnls of vector £? b. Draw a coord inate system and on it show vec tors A, B,

and E. C. What arc the magnitude and direction of vector £? II For the three vectors shown in

Figure P3.45 , the vector sum jj = if + jj + C has components D" = 2 and 0 ,. = O.

y

4 A ----='i-::~~>- x

a. What are the x- and y-components 2 of vector 8? C

b. Write jj as a magn itude and a

direction. FIGURE P3.45

46. II Let A = (3.0 m. 20° south of east). jj = (2.0 m. nonh), and C = (5.0 m. 70° south of west). a. Draw and label A. ii, and C with the ir tail s at the origin . Use

a coordinate system with the x-axis to the east. b. Write thex- and y-components of vectors A, 8, and C. c. Find the magnitude and the direction of jj = A + jj + C.

47. II A typical set of stairs is angled al 38°. You c limb a se t of stairs at a speed of 3.5 m/s. a. How much height will you gain in 2.0 s? b. How much hori zontal distance will you cover in 2.0 s?

48. II The minute hand on a watch is 2.0 cm long. What is the di s­placement vector of the tip of the minute hand a. From 8:00 to 8:20 A.M.? b. From 8:00 to 9:00 A.M.?

49. 11111 A fie ld mouse trying to escape a hawk runs east for 5.0 m, darts southeast for 3 .0 m, then drops 1.0 m down a hole into its bu rrow. What is the magnitude of the net di splacement of the mouse?

50. AI A pilot in a small plane encounters shifting winds. He nies 26.0 km northeast, then 45.0 km due north . From thi s point. he nies an addi ti ona l di stance in an unknown direc tion , only to find himself at a small airslfip that hi s map shows to be 70.0 km direc tly north of hi s starting point. What were the length and direction of the third leg of hi s trip?

5 1. II A small plane is 100 km south of the eq uator. The plane is flying at 150 km/h at a heading of 300 to the west of north . In how many minutes will the plane cross the equator?

52. II The bac teriulll Escherichia coli (or E. coli) is a single­BID celled organism that li ves in the gut of hea lthy humans and

animals . When grown in a uniform medium ri c h in sa lt s and amino acids, these bacteria swim along z ig-zag paths at a constan t speed of 20 p,m/s. Figure P3.52 shows the trajectory o f an E. coli as it moves from point A to point E. Each seg­ment of the motion can be identified by two letters, s llc h as segme nt Be. a. For each of the four segments in the bac terium 's trajectory.

calculate the x- and y-components of its di splacement and of its veloc ity.

b. Calculate both the total di stance traveled and the magnitude of the net di splacement for the entire mot ion.

c. What are the magnitude and the direction of the bacterium's average velocity for the en tire trip?

53.

54.

Problems 99

y (p.m )

40,------------------------,

30 o

20

10 A or"", 0:=-:2"0:-:30';:--40';:--:' "0" 60(:--=7"0 ---='SOC:-"'90=-'c!OO x (p. m)

- '0

- 20 E

- 30

-40 L-----------------------~

FIGURE P3.52

1111 A skier is gliding along at 3.0 mls on horizontal , frictionless snow. He suddenly starts down a 10° incline. His 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? III A block slides a long the frictionless track shown in Figure P3.54 with an ini tial speed of 5.0 m/s. Assume it turns aU the corners smoothly, with no loss of speed.

~.o m 45° 30g

1--------01 - - - - - - - -

I.Om

FIGURE P3 .54

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

s ide? c. By what percentage does the block's final speed differ from

its initial speed? Is thi s surpri s ing? 55. III One game at the amuse,ment park has you push a puck up a

long, frictionless ramp. You win a stuffed animal if the puck, at its highest point, comes to within 10 cm of 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 moving side­walk the entire way, without wa lking, it takes 75 s to travel (he same di stance. How long will it take YOll 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. For the nex t two hours, ship A travels al 20 mph in a direc tion 30° west of north while ship B trave ls 20° cast of north at 25 mph. a. What is the di s tance betwee n the two ships 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 lOO-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 isplacement due to rowing.

her di splacement due to the river's motion, and her net d is­placement.

59. III A nock of ducks is try ing to migrate south for the winte r, but they keep being blown 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 angle to the wind. If the ducks can fly at 16 mls relat ive (0 the air, in what direction should they head in order to move directly south ?

100 CHAPTER 3 Vectors and Motio n in Two Dimen s io ns

60. III A kayaker needs to paddle north across a 1 OO- m-w ide har­bor. T he tide is go ing ou t, creat ing a tidal current nowing east at 2.0 m/s. The kayaker can paddle with a speed of 3.0 mls. a. In which d irection should 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 destinat ion 600 mi due east, but a wind is blowi ng at 50 mph in the d irec tion 30° north of east. a. In what d irect ion must the p ilot head the plane in order to

reach her desti nation? b. How long will the tri p take?

62. III The Gulf S tream off the eas t coast of 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 of 10 m/s. T he capta in wo uld li ke to reach la nd at a po int due west fro m the current pos ition. a. In what direction 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 Planet y

Exidor throws a ball , and it fol­lows the parabolic trajectory shown in Figure P3.63. The ball 's position is shown at 1.0 s in ter-va ls un til { = 3.0 s. At { = 1.0 s,

l .Os

1.0 s 3.0s

the baU's veloc ity has compo- ol<,----------x

nents Vx = 2.0 mIs, II.\" = 2.0 m/s. FIGURE Pl .6l

a. Determine the x- and y­

components of the baU's velocity at 1 = 0.0 s, 2.0 s, and 3.0 s. b. What is the value of g on Planet Ex idor? c. What was the ball's lau nch angle?

64. III A ball thrown horizontall y at 25 mls travels a horizontal d is­ta nce of 50 m before hi tt ing the ground. From what height was the bal l throw n?

65. III In 1780, in what is now referred BIO to as "Brady's Leap," Captain Sam

Brady of the U.S. Cont inental Army escaped certa in death from hi s enemies by run ning over the edge of the cl iff above Ohio 's h

Cuya hoga River, which is con-fin ed at that spot to a gorge. He landed safely on the far s ide of the ri ver. It was reported that he leapt 22 ft across while fa lling 20 ft. FIGURE Pl .6S

L

a. Representing the d istance jumped as L and the vertical drop as II , as shown in Figure P3.65, deri ve an express ion for the minimum speed )I he would need to make hi s leap if he ran straight off the cl iff.

b. Eval uate yo ur express ion for a 22 ft j um p with a 20 ft drop to the other s ide.

c . Is it reasonable that a person could make thi s leap? Use the fact that the worl d record for the 100 m dash is approx i­mately 10 s to est imate the max imum speed such a run ner would have.

66. III The longest recorded pass in an NFL game traveled 83 yards in the ai.r from the quarterback to the rece iver. Assu ming that the pass was thrown at the optimal 45° angle, what was the speed at which the ball le ft the quarterback's hand?

67. III A spring- loaded gun , fi red vert icall y, shoots a marble 6.0 m straight up in the air. What is the marble 's range if it is fi red hor­izon tall y from 1.5 m above the ground?

68. II In a shot-put event , an athlete throws the shot with an ini tial speed of 12.0 mls at a 40.0° angle from the hori zon tal. The shot leaves her hand at a height of 1.80 m above the ground. a. How far does the shot travel? b. Repeat the calcul at ion of part (a) for angles 42.5°, 45.0°, and

47.5°. Put a ll your results, includ ing 40.0°, in a table. At what angle of release does she throw the fanhest?

69. III A tenni s player hits a ball 2.0 m above the ground. The ball leaves hi s racq uet with a speed of 20 mls at an ang le 5.00 above the horizonta l. The hor izontal d istance to the net is 7.0 m, and the net is 1.0 m high. Does the ball clear the net? If so, by how much? If not, by how much does it miss?

70. 1111 Water at the top of Horseshoe Fall s (part of Niagara Falls) is moving horizontall y at 9.0 mls as it goes off the edge and plunges 53 m to the pool be low. If you ignore air res ista nce, at what angle is the fa lling water moving as it en ters the pool?

7 1. III Figure 3.37 shows that the range of a projecti le launched at a 60° angle has the same range as a projec til e launched at a 300

angle-but they wo n't be in the air for the same amoun t of time. Suppose a project ile launched at a 30° angle is in the air fo r 2.0 s. How long will the projectile be in the air if it is launched with the same speed at a 60° angle?

72. II A supply plane needs to drop a package of food to sc ient ists working on a glac ier in Greenland. T he plane flies J 00 m above the glac ier at a speed of 150 m/s. How far short of the target should it drop the package?

73. III A chi ld sl ides down a frict ionless 3.0-m-long playground sl ide tilted upward at an a ngle of 40°. At the e nd of the sl ide, there is an add itional section that curves so that the child is launched off the end of the slide horizontally. a. How fast is the child moving at the bottom of the sl ide? b. If the end of the slide is OAO III above the ground, how far

from the end does she land? 74. III A sports car is advertised to be able to "reach 60 mph in 5 sec­INT onds flat, comer at 0.85g, and stop from 70 mph in only 168 fee t."

a. In whic h of those three sit uat ions is the magni tude of the car's accele rati on the largest? In which is it the smallest?

b. At 60 mph, what is the smallest turni ng radius that thi s car can nav igate?

75. A Ford Mustang can accelerate from 0 to 60 mph in a time of INT 5.6 s. A Mini Cooper isn' t capable of such a rapid start, but it

can tu m in a very small circle 34 ft in d iameter. How fast would yo u need to dri ve the Mini Cooper in thi s tight c ircle to match the magni tude of the Mustang's accelerati on?

76. II The "Scream ing Swing" is a carnival ri de that is-not surpris ingly- a giant swing. It's actually two swings moving in oppos ite d irect ions. At the bottom of its arc, riders are moving at 30 mls with respect to the ground in a 50-m-diameter circle. a. What is the accelerat ion, in m/s2 and in units of g, that riders

experience? b. At the bottom of the ride, as they pass each othe r, how fast

do the riders move with respect to each other? 77. On an otherwise straight stretch of road near Moffat, Colorado,

the road suddenl y tu rns. This bend in the road is a segment of a circle with radius 110m. Drivers are cautioned to slow down to 40 mph as they navigate the curve. a. If you heed the sign and slow to 40 mph, what wi ll be your

acce lerati on go ing around the curve at thi s constant speed? Give your answer in m/s 2 and in units of g.

b. At what speed would your accelerat ion be double that at the recomme nded speed?

Passage Problems

Riding the Water Slide

A rider on a water s lide goes through three di fferent kinds of mot ion, as illustrated in Figure P3.78. Use the data and details from the fi g­ure to answer the fo llowing questions.

1. TIle tirst section of the motion is a ramp with no friction: riders ~laT1 at re~t and ac:cderJle down the r;lmp.

2. The ,",ood ~"ioo of'h' // I motion i~ a circular segment that change, the direction of r = [ 5 III mOlion: riders go ;lround this . circular segment at a const:mt __ ~ h = 0.00 III speed and end with a velocity ___ _ --'-.

that is horizontal. 'f" '''_

3. The third ,ection orlhe motion j, a parabolic tr.ljectory .. "" through lhe air at the end of which riders land in the waler. ",

FIGURE P3 .78

Stop to Think 3.1: A. The graphica l construc­tion of 2? - Q is shown at right.

Stop to Think 3.2: From the axes on the graph, we can see that the x- and y-components are -4 cm and +2 cm, respect ively.

Stop to Think 3 .3: C. VeClOr C poin ts to the left

Problems 101

78. I At the e nd of the first sect ion of the motion, riders are mov­ing at what approx imate speed? A. 3 m/s B. 6 m/s C. 9 m/s D. 12 m/s

79. I Suppose the accelerat ion during the second sect ion of the mot ion is too large to be comfortab le for riders. What change could be made to dec rease the acce lerati on duri ng thi s sect ion? A. Reduce the rad ius of the c ircular segment. B. Increase the rad ius of the circular segmen t. C. Increase the ang le of the ramp. D. Increase the le ngth of the ramp.

80. I What is the vertical component of the velocity of a rider as he or she hi ts the water? A. 2.4 m/s 8. 3.4 mls C. 5.2 mls D. 9.1 mls

81. I Suppose the des igners of the water sl ide want to adj ust the height" above the water so that ri ders land twice as far away from the bottom of the sl ide. What would be the necessary he ight above the water? A. 1.2 m B. 1.8 m C. 2.4 m D. 3.0m

82. I During which section of the mo tion is the mag ni tude of the acce lerat ion experienced by a rider the greatest? A. The firs t. B. The second. C. T he thi rd . D. It is the same in all sect ions.

S top to Think 3.4: B. The angle of the s lope is greatest in this case, lead ing to the greatest accelerat ion.

S top to Think 3.5: D. Mass does not appear in the kinemat ic equa­tions, so the mass has no effect; the ball s will fo llow the same path.

Stop to Think 3.6: B. T he magn itude of the acce lerat ion is v2/r .

Acce leration is largest for the combination of highest speed and and dow n, so bo th c.. and C,. are negat ive. C. is in the numerator smallest rad ius. because it is the side opposite cb.