One car travels due east at and a second car travels north at Are their velo0wxm k+w,r( mzcit+0,(w xwk mmzries equal? Explain.

Can you give several examples of an object’s motion in which a grla tdb/ aa9 )sr2)1ser. ffgv gg*6c+ceat distance is traveled but the displacement is zs21rgabs 6)tv*aa9gfcc) f/+ .g r eldero?

Can the displacement vector for a particp *en2vsoe2igy -gg 2f/eql(4le moving in two dimensions ever be longer than the length of path traveled by the particle over the same time interval? Cag-*q4e g(py2vefoil/g sn2e 2n it ever be less? Discuss.

$\text { If } \overrightarrow{\mathbf{V}}=\overrightarrow{\mathbf{V}}_{1}+\overrightarrow{\mathbf{v}}_{2} \text {, is } V \text { necessarily greater than } V_{1} \text { and/or } V_{2} \text { ? Discuss. }$

  Two vectors have lengthbon+-o .ukh :q $ V_{1}=3.5 \mathrm{~km} $ and $ V_{2}=4.0 \mathrm{~km}$ . What are the maximum and minimum magnitudes of their vector sum?max =    min =   

Can two vectors of unequal magnitude add up to give the zero vector? -m, z+ 7irgm,w un5xjrCan three unequal vnmrw g r xi+m,-5zuj7,ectors? Under what conditions?

Can the magnitude of a vector ever (a) be equal to one37q*tz i371u oyndwra1d. j lv of its components, or (b) bedyzt.vo*17uwj73in3 lr q 1 da less than one of its components?

  Can a particle with constant speed be accelerating? ----待选择----YesNo  What if it has constant velocity? ----待选择----YesNo 

At some amusement parks, to getxg ib5e3 ;t;yl on a moving “car” the riders first hop onto a moving walkway and the;l gbe t5yxi3;n onto the cars themselves. Why is this done?

If you are riding on a train that speeds past another tram,mws/.evlc 3 ltk25v in moving in the same directe/ 35c ,lk wsvm.2lmvtion on an adjacent track, it appears that the other train is moving backward. Why?

If you stand motionless undnq-a+u/ 3iwpht 2t3jzer an umbrella in a rainstorm where the drops fall verti+2uhtai pt/wq jz3-3ncally, you remain relatively dry. However, if you start running, the rain begins to hit your legs even if they remain under the umbrella. Why?

  A person sitting in an enclosed train car, movinglo5i u2 grii 97m 6nwu*qp-bh1 at constant velocity, throws a ball straight up into the air in her reference frabh9 li*wgp-n5iqm 2ui7ro1 u6me.
(a) Where does the ball land? What is your answer if the car  ----待选择----ahead of herbehind herin her handsher left 
(b) accelerates,  ----待选择----ahead of herbehind herin her handsher left 
(c) decelerates,  ----待选择----ahead of herbehind herin her handsher left 
(d) rounds a curve,  ----待选择----ahead of herbehind herin her handsher left 
(e) moves with constant velocity but is open to the air?  ----待选择----ahead of herbehind herin her handsher left 

  A and B rowers, who can row at the same speed in still water, set off across ,5.sx b pnd n0d,sqf(za river at the same time. A heads straight across and is pulled downstream somewhat by the current. The B heads upstream at an angle so as to arrive at a point opposite the starting point. Which rower reaches the opp0s5nx, z.dfb(n qd,sposite side first? ----待选择----AB 

How do you think a basebaka.h kkac3 5i.ll player “judges” the flight of a fly ball? Which equation in this Chapter becomes part of t hka. k3i.ack5he player’s intuition?

In archery, should the arrow be aimed directly at the tayhiz* 5b8vso8rget? How should your angle of aim depend on the distance*i5 ob ysh8z8v to the target?

  A projectile is launched at an 4( s5- ;z zjqxe/0x/--vommfyylckyo angle of $30^{\circ}$ to the horizontal with a speed of $ 30 \mathrm{~m} / \mathrm{s}$. How does the horizontal component of its velocity 1.0 s after launch compare with its horizontal component of velocity 2.0 s after launch?   

  Two cannonballs, A and B , are fired l5ode-;d1 pgu from the ground with identical initial speee;gu l -dp1od5ds, but with $ \theta_{\mathrm{A}} $ larger than $ \theta_{\mathrm{B}} $
(a)Which cannonball reaches a higher elevation? ----待选择----AB 
(b) Which stays longer in the air?  ----待选择----AB 
(c) Which travels farther? If the angle of a is 40 $^{\circ}$ the angle of b is 30$^{\circ}$  ----待选择----AB 

A car is driven 215 jilj2 4g.v2pjd os./jkm west and then 85 km southwest. What is the displacement of the car from the point of origpo2ds gj2j/.vlj4.ji in (magnitude and direction)? Draw a diagram.

  A delivery truck travels 18 blocks nwzyt+ 3 97 :/sehep97jxh hsrqorth, 10 blocks east, and 16 blocks south. What is its final displacement from the origin? Assume the blocks are equal tzs79qe e w7/h93s :r xyhhj+plength.    blocks

$\text { Show that the vector labeled "incorrect" in Fig. 3-6c is actually the difference of the two vectors. Is it }\\
\overrightarrow{\mathbf{v}}_{2}-\overrightarrow{\mathbf{v}}_{1} \text {, or } \overrightarrow{\mathbf{v}}_{\mathbf{1}}-\overrightarrow{\mathbf{v}}_{\mathbf{2}} ?$

  $\text { If } V_{x}=6.80 \text { units and } V_{y}=-7.40 \text { units, determine the magnitude and direction of } \overrightarrow{\mathbf{V}} \text {. }$ magnitude    units direction    $^{\circ}$

Graphically determine5 7gaqgd/ opc--: 2 m3zw huzrqv02v,kjbzc;w the resultant of the following three vector displacements: gzk -hcvr2m 0/zpv q c35g7u-;w,o daz2 bq:wj(1) $ 34 \mathrm{~m}, 25^{\circ} $ north of east; (2) $48 \mathrm{~m}, 33^{\circ}$ east of north; and (3) $22 \mathrm{~m}, 56^{\circ} $ west of south.

  The components of a ve zxv+e-ne6v )ge-2vg(z r,kdtctor $\overrightarrow{\mathbf{V}}$ can be written $ \left(V_{x}, V_{y}, V_{z}\right)$ . What are the components and length of a vector which is the sum of the two vectors, $ \overrightarrow{\mathbf{V}}_{1} $ and $\overrightarrow{\mathbf{V}}_{2}$ , whose components are (8.0,-3.7,0.0) and (3.9,-8.1,-4.4) ? components (a,b,c) a =    b =    c =   
length   

  $\overrightarrow{\mathbf{V}}$ is a vector 14.3 units in magnitude and points at an angle of $34.8^{\circ} $ above the negative x axis.
(a) Sketch this vector.
(b) Find $V_{x}$ and $V_{y}$ .$V_{x}$ =    $V_{y}$ =   
(c) Use $ V_{x}$ and $ V_{y}$ to obtain (again) the magnitude and direction of $\overrightarrow{\mathbf{V}}$ . magnitude =    direction =    $^{\circ}$ above - the x axis
[Note: Part (c) is a good way to check if you've resolved your vector correctly.]

  Vector $\overrightarrow{\mathbf{V}}_{1}$ is 6.6 units long and points along the negative x axis. Vector $\overrightarrow{\mathbf{V}}_{2} $ is 8.5 units long and points at $+45^{\circ}$ to the positive x axis. (a) What are the x and y components of each vector? (b) Determine the sum $\overrightarrow{\mathbf{V}}_{1}+\overrightarrow{\mathbf{V}}_{2}$ (magnitude and angle)

$ (a) V_{1 x}$=   units $V_{1 y}$=    units $V_{2 x}$=    units $\quad V_{2 y}$=    units


$\left|\overrightarrow{\mathbf{V}}_{1}+\overrightarrow{\mathbf{V}}_{2}\right|$=   units $ \quad \theta$=  $^{\circ}$

  An airplane is traveli:th k oaek4(1wng $735 \mathrm{~km} / \mathrm{h}$ in a direction $ 41.5^{\circ} $ west of north (Fig. 3-31). (a) Find the components of the velocity vector in the northerly and westerly directions. (b) How far north and how far west has the plane traveled after 3.00 h ?(a ) $ v_{\text {north }} $=   $v_{\text {west }} $=    (b) $\Delta d_{\text {north}} $=    $\Delta d_{\text {west}}$ =   

  Three vectors are shown in Fig. below. Their magnisdn)m;d27bt0is57qua/ v * bfzmpi5ktudes are given in arbitrary units. Determine the sum of the three vectors. Give the resultant in terms of (a) components, (b) magnitude and a b );q5tv7b/2f5 a mk*nss0dup 7zmidingle with the x axis.


$(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}})_{x}$ =   
$(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}})_{y}$ =   
$|\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}|$ =   
$\theta$ =    $^{\circ}$

  $\text { Determine the vector } \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{C}} \text {, given the vectors } \overrightarrow{\mathbf{A}} \text { and } \overrightarrow{\mathbf{C}} \text { in Fig. below}$



$|\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{C}}|$ =   
$\theta$ =    $^{\circ}$

  (a) Given the vectors;b-wf. qpln2 h $ \overrightarrow{\mathbf{A}}$ and $ \overrightarrow{\mathbf{B}}$ shown in Fig. below , determine $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}$ . (b) Determine $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ without using your answer in (a) . Then compare your results and see if they are opposite.


$|\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}|$ =   
$\theta_{B-A}$ =    $^{\circ}$above -x axis
$|\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}|$ =   
$\theta$ =    $^{\circ }\text { below }+x \text { axis }$

  $\text { For the vectors given in Fig. 3-32, determine (a) } \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}} \text {, (b) } \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{C}} \text {, and (c) } \overrightarrow{\mathbf{C}}-\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} \text {. }$




$|\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}|$ =   
$\theta$ =    $^{\circ}$ below +x axis
$|\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{C}}|$ =   
$\theta$ =    $^{\circ}$
$|\overrightarrow{\mathbf{C}}-\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}|$ =   
$\theta$ =    $^{\circ}$ below - x axis

  $\text { For the vectors shown in Fig. below, determine (a) } \overrightarrow{\mathbf{B}}-2 \overrightarrow{\mathbf{A}} \text {, (b) } 2 \overrightarrow{\mathbf{A}}-3 \overrightarrow{\mathbf{B}}-2 \overrightarrow{\mathbf{C}} \text {. }$



$|\overrightarrow{\mathbf{B}}-2 \overrightarrow{\mathbf{A}}|$ =   
$\theta$ =    $^{\circ}$ below - x axis
$|2 \overrightarrow{\mathbf{A}}-3 \overrightarrow{\mathbf{B}}+2 \overrightarrow{\mathbf{C}}|$ =   
$\theta$ =    $^{\circ}$ below + x axis

  The summit of a mountain, 2450 m above base camp, is measured on a map to 20b1m3sx-- ldq jcgo xbe 4580 m horizontally fro-bxq olgd m xj-30s21cm the camp in a direction $32.5^{\circ}$ west of north. What are the components of the displacement vector from camp to summit? What is its magnitude? Choose the x axis east, y axis north, and z axis up. x =    y =    $|\overrightarrow{\mathbf{r}}|$ =   

  You are given a vector in the xy plane that has a magnit* u;r bqftuup1b6k o,0ude of 70.0 units and a yb;k1 u*q rf u,tubo06p component of What are the two possibilities for its x component? x = $\pm$   

  A tiger leaps horizontally from a 6.5-m-high rock with a speed of How fewcp)u+m u)+m6 .gb aoar from the ba)p.mugb ++cwmo) 6aeuse of the rock will she land?    m

  A diver running dives out horizontally fro0o u 2iop.oah id1h1+k:2rfyhm the edge of a vertical cliff and 3.0 s1i k. +ohpy:oir1 uhafd02h2o later reaches the water below. How high was the cliff, and how far from its base did the diver hit the water? high of cliff is    the distance is   

A fire hose held near the groue b sm ieo 0p4 v9*1magdn40scu,jt(n8nd shoots water at a speed of At what angle(s) should the nozzle point in order that the water land 2.0 m away (Fig. below)? Why are there two different angles? Sketch the two tras4 4ng d9,n10acjmeem8p0 b tvosi*(u jectories.

  Romeo is chucking pebbles gently up ;yfhtj( )r1qdj7;nb o to Juliet’s window, and he wants the pebbles to hit the window with only a horizontal component of velocity. He is standing at the edge of a rose garden 4.5 m below her window and 5.0 m fro;q jj7 h1dyfnb;( to)rm the base of the wall (Fig. below). How fast are the pebbles going when they hit her window?    m/s

  A ball is thrown horiz3z81p)q 8kdok, i. jibnhltw3ontally from the roof of a building 45.0 m tall and lands 24.0 m from the base. What was the ball’s initial speel.,wpkd ijt81bno) kh3i3q8z d?   

  A football is kicked at ground level with a speed tlkltzsg 6g vk+u+h(3 83f/h xof 18.0 $\mathrm{~m} / \mathrm{s} $ at an angle of 35.0$^{\circ}$ to the horizontal. How much later does it hit the ground?    s

  A ball thrown horizonu ;b7 bp/jnjm4tally at from the roof of a building lands 36.0 m from the base of n;7 u jm4pbjb/the building. How tall is the building?    m

  An athlete executing a loncex 4nt1)sh09iqefc* g jump leaves the ground at a 28.0$^{\circ}$ angle and travels 7.80 m .
(a) What was the takeoff speed?   
(b) If this speed were increased by just 5.0 % , how much longer would the jump be?   

  Determine how much farther a person can jui 8 m 6ex6v7ilivi4g (,h/gnmpmp on the Moon as compared to the Earth if the takeoff speed and a vmh6g6xvii48m g pl/,in 7ei(ngle are the same. The acceleration due to gravity on the Moon is one-sixth what it is on Earth.    times

  A hunter aims directu hj yi1b1e :/wu0ed;hly at a target (on the same level) 75.0 m away.
(a) If the bullet leaves the gun at a speed of by how much will it miss the target?  
(b) At what angle should the gun be aimed so as to hit the target?    $^{\circ}$

  The pilot of an airplane traveling wante8rjlmzm9;w 2 s to drop supplies to flood victims isolated mjl2wezr;9m8 on a patch of land 160 m below. The supplies should be dropped how many seconds before the plane is directly overhead?    s

Show that the speed with which a projectile leaves the ground is equal to its speed just before it strikes the ground at the end of its journey, assuming the firing level equals the landing level.

  A projectile is fired with an initial lt6qng)6bp )s+ xqb7d speed of at an angle of 34.5$^{\circ}$ above the horizontal on a long flat firing range. Determine
(a) the maximum height reached by the projectile,    m
(b) the total time in the air,    s
(c) the total horizontal distance covered (that is, the range), and    m
(d) the velocity of the projectile 1.50 s after firing.    m/s

  A projectile is shot from the edge of a cliff 125 m above ground level wit/);/d xbie (f/jtz v*5lkcrxi h an initial speed of at an angljtzk/ f(lv b; ci r)x5de/ix*/e of 37.0º with the horizontal, as shown in Fig. below.
(a) Determine the time taken by the projectile to hit point P at ground level.   
(b) Determine the range X of the projectile as measured from the base of the cliff. At the instant just before the projectile hits point P, find   
(c) the horizontal and the vertical components of its velocity,   
(d) the magnitude of the velocity, and   
(e) the angle made by the velocity vector with the horizontal.    $^{\circ}$ below the horizon
(f) Find the maximum height above the cliff top reached by the projectile.   

  A shotputter throws the shot with an initial*yt 2fslap qxj -1t:(tf;ir 3i speed of 15.5 $\mathrm{~m} / \mathrm{s} $ at a $ 34.0^{\circ}$ angle to the horizontal. Calculate the horizontal distance traveled by the shot if it leaves the athlete's hand at a height of 2.20 m above the ground.   

  At what projection angle will the range of a projectile equal its maximy+7oyijn-v6 92/nncly na1c aum height? nn6c iaojcn9v 2n-7ayl /y+y1   ${\circ}$

Revisit Conceptual Example 3-7, and assume that the boy pj 0f -rjx./howith the slingshot is below the boy in the tree (Fig. 3-36), and so aims upward, directly at the boy in the tree. Show that again the boy in the tree makes th xp 0j.fj-/hore wrong move by letting go at the moment the water balloon is shot.

  A rescue plane wants to drop supplies to isolated mountain climbers on+a )4iuwftt5mb-sio g* mi 2v/v//;j4a stfk n a rocky ridge 24i*t j/v4 tvnmg ba5fa-s/wo m2 ii k/u+;)fst35 m below. If the plane is traveling horizontally with a speed of 250 $\mathrm{~km} / \mathrm{h}(69.4 \mathrm{~m} / \mathrm{s})$

(a) how far in advance of the recipients (horizontal distance) must the goods be dropped (Fig. a)?    m
(b) Suppose, instead, that the plane releases the supplies a horizontal distance of 425 m in advance of the mountain climbers. What vertical velocity (up or down)  ----待选择----updown  should the supplies be given so that they arrive precisely at the climbers' position (Fig. b)?    m/s (fill in magnitude here)
(c) With what speed do the supplies land in the latter case?    m/s

  A person going for a morn 8 i8o xg5naflm (xu-jo*6(onming jog on the deck of a cruise ship is running toward the bow (front) gmf joi86x8 -ln 5nam(o(u*oxof the ship at 2.2 $\mathrm{~m} / \mathrm{s}$ while the ship is moving ahead at 7.5 $\mathrm{~m} / \mathrm{s}$ . What is the velocity of the jogger relative to the water?    m/s
Later, the jogger is moving toward the stern (rear) of the ship. What is the jogger's velocity relative to the water now?    m/s

  Huck Finn walks at a speed o7 t(frs/q/8 ondzy,cpf 0.60$ \mathrm{~m} / \mathrm{s}$ across his raft (that is, he walks perpendicular to the raft's motion relative to the shore). The raft is traveling down the Mississippi River at a speed of 1.70$ \mathrm{~m} / \mathrm{s}$ relative to the river bank (Fig. 3-38). What is Huck's velocity (speed and direction) relative to the river bank?Magnitude =    Direction =    $^{\circ}$

  You are driving south on a highwaswl jwd(*ry26x/g p 2ay at 25 $\mathrm{~m} / \mathrm{s} $ (approximately 55$ \mathrm{mi} / \mathrm{h} ) $in a snowstorm. When you last stopped, you noticed that the snow was coming down vertically, but it is passing the windows of the moving car at an angle of $30^{\circ} $ to the horizontal. Estimate the speed of the snowflakes relative to the car    and relative to the ground.   

  A boat can travel 2.30 whpcb z(::eu228ib*vg3d: dbuubfk: $ \mathrm{~m} / \mathrm{s}$ in still water.
(a) If the boat points its prow directly across a stream whose current is 1.20$ \mathrm{~m} / \mathrm{s}$ , what is the velocity (magnitude and direction) of the boat relative to the shore? magnitude    direction    $^{\circ}$
(b) What will be the position of the boat, relative to its point of origin, after 3.00 s? (See Fig. below)diraction =    awayfrom the starting point,angle =    $^{\circ}$ relative to the shore

  Two planes approach each other head-on. Each has a s. cj btp(;mdgtobh9b/- ;o+li j;b)y ypeed of and they spot each other when they are initially 1ol.g;m(b +;jb9 hijpt; b oy) t-cy/bd1.0 km apart. How much time do the pilots have to take evasive action?    s

  An airplane is heading due south at cj;m1j9fgk era(rd1 5+v mh w)a speed of 600$ \mathrm{~km} / \mathrm{h} $. If a wind begins blowing from the southwest at a speed of 100 $\mathrm{~km} / \mathrm{h}$ (average), calculate:
(a) the velocity (magnitude and direction) of the plane relative to the ground, and    $^{\circ} $ east of south
(b) how far from its intended position will it be after 10 min if the pilot takes no corrective action.   
[Hint: First draw a diagram.]

  In what direction should the pilot aim the plane in P85e o1 +t2 pi(l+ sqlajl9p crn7j;lhnroblem 41 so that it will fly due southr 2+i ph+cl(n qolnp9;5tj1alj 7e sl8?    $^{\circ} $ east of south

  Determine the speed of the boat with re -risril rlfdd-26,wft it-/*spect to the shore in Example 3–11.   

  A passenger on a boat moving at 1oi n jq1z* 66cjscgm3 1.50$ \mathrm{~m} / \mathrm{s}$ on a still lake walks up a flight of stairs at a speed of 0.50 $\mathrm{~m} / \mathrm{s}$ (Fig. below). The stairs are angled at $45^{\circ}$ pointing in the direction of motion as shown. What is the velocity of the passenger relative to the water?   

  A motorboat whose speed in st+vmyche(2- .fn * wvuzau2/5p empyt-ill water is 2.60 $\mathrm{~m} / \mathrm{s}$ must aim upstream at an angle of 28.5$^{\circ}$ (with respect to a line perpendicular to the shore) in order to travel directly across the stream. (a) What is the speed of the current?   
(b) What is the resultant speed of the boat with respect to the shore?   

  A boat, whose speed in stsc11.ia0x )s/z o xbffill water is 1.70 $\mathrm{~m} / \mathrm{s} $, must cross a $260-\mathrm{m}$ -wide river and arrive at a point 110 m upstream from where it starts (Fig. below). To do so, the pilot must head the boat at a 45$^{\circ}$ upstream angle.


What is the speed of the river's current?   

  A swimmer is capable of swimm4hkg.8aki ehs5:xe5 ding 0.45 $\mathrm{~m} / \mathrm{s} $ in still water. (a) If she aims her body directly across a 75 -
m -wide river whose current is 0.40 $\mathrm{~m} / \mathrm{s}$ , how far downstream (from a point opposite her starting point)
will she land?   
(b) How long will it take her to reach the other side?   

  A swimmer is capable of swimming 0d ddyo/a69s8p .45 $\mathrm{~m} / \mathrm{s} $ in still water. (a) If she aims her body directly across a 75 -
m -wide river whose current is 0.40 $\mathrm{~m} / \mathrm{s}$ , how far downstream (from a point opposite her starting point)
will she land?    $^{\circ} $
(b) How long will it take her to reach the other side?   

  An airplane whose air speeo1juo7(tguv2*i ;tb td is 620$ \mathrm{~km} / \mathrm{h}$ is supposed to fly in a straight path 35.0$^{\circ}$ north of east. But a steady 95$ \mathrm{~km} / \mathrm{h} $ wind is blowing from the north. In what direction should the plane head?    $^{\circ} $ north of east

  An unmarked police car, traveling a cond:2eb qh:u hr*stant 95$ \mathrm{~km} / \mathrm{h}$ , is passed by a speeder traveling 145$ \mathrm{~km} / \mathrm{h}$ .

Precisely 1.00 s after the speeder passes, the policeman steps on the accelerator. If the police car's acceleration is 2.00 $\mathrm{~m} / \mathrm{s}^{2}$, how much time elapses after the police car is passed until it overtakes the speeder (assumed moving at constant speed)?   

  Assume in Problem 50 that the speeder’s speedi;wv5uj16495n znmi ude m:64zm fees is not known. If the police car accelerates uniformly6e is z6e9fen1 n4iu:mv4m m5;wjud5z as given above, and overtakes the speeder after 7.00 s, what was the speeder’s speed?    km/h

  Two cars approach a street corner at right angle37exyz.c1 0qc2qk sczs to each other (Fig. below). Car 1 travels at1s3kxz0zq2qec c .yc7 a speed relative to Earth $v_{1 \mathrm{E}}=35 \mathrm{~km} / \mathrm{h}$ , and car 2 at $v_{2 \mathrm{E}}=55 \mathrm{~km} / \mathrm{h}$ . What is the relative velocity of car 1 as seen by car 2 ?    km/h    $\theta$$^{\circ} $ West of North
What is the velocity of car 2 relative to car 1 ?    km/h    $\theta$ $^{\circ} $ south of east

  William Tell must split the apple atop his son's head frdr t+k3 j*vop*r+ee5rom a distance of 27 m . When William aims directly at the apple, the arrow is horizertdv3 o5p*rje*+ k+rontal. At what angle must he aim it to hit the apple if the arrow travels at a speed of 35$ \mathrm{~m} / \mathrm{s}$ ?    $^{\circ} $

  A plumber steps out of his truck, walks 50 m east and 25 m south, and then takesoai f)5 -et*et an elevator 10 m down into the basement of a building where a bad leak is occurring. What is the displacement of the plumber relative to his truck? Give your answeratfi *t5 ee-o) in components, and also give the magnitude and angles with the x axis in the vertical and horizontal planes. Assume x is east, y is north, and z is up.

The result is that the displacement is    m , at an angle of    $^{\circ}$ South of East, and    $^{\circ}$ Below the Horizontal.

  On mountainous downhill roads, escape routes are sometjvxya );h;,qc imes placed to the side of the road for trucks whose brakes mi cq),; ja;xhyvght fail. Assuming a constant upward slope of 32$^{\circ}$ , calculate the horizontal and vertical components of the acceleration of a truck that slowed from 120 $\mathrm{~km} / \mathrm{h}$ to rest in 6.0 s . See Fig. below

$\text { The horizontal acceleration is } a_{\text {hoiz }}$ =    $\mathrm{m} / \mathrm{s}^{2}$
$\text { The vertical acceleration is } a_{\text {vert }}$ =    $\mathrm{m} / \mathrm{s}^{2}$

  What is the y component of a vector (in the x y +n j7wi pq +fm53snml7 plane) whose magnitude is 88.5 and who5p 3fw+l7q+ i jns7mnmse x component is
75.4? What is the direction of this vector (angle it makes with the x axis)?
$\text { Direction }$=    $^{\circ}$$ \text { relative to } x \text { axis }$

  Raindrops make an angle e(y ;xath(k , t)eo;:pz 4c:8i pwlghh0uvf7 p $\theta=30^{\circ}$ with the vertical when viewed through a moving train window (Fig. below). If the speed of the train is $20 \; m/s$ (a) what is the speed of the raindrops in the reference frame of the Earth in which they are assumed to fall vertically?    $m/s$
(b) what is the speed of the raindrops relative to the train?    $m/s$

  A light plane is headed due sout6 wnii5kav9*6islf7u d,hs grd.x5t2h with a speed of 155$ \mathrm{~km} / \mathrm{h}$ relative to still air. After 1.00 hour, the pilot notices that they have covered only 125 km and their direction is not south but southeast $\left(45.0^{\circ}\right) $. What is the wind velocity? $v_{wind}$ =    km/h $\text { The direction of the wind is } \theta$=    $^{\circ} $ north of east

A car moving at 95 +jyjk s4j49ik $\mathrm{~km} / \mathrm{h}$ passes a 1.00$-\mathrm{km} $-long train traveling in the same direction on a track that is parallel to the road. If the speed of the train is 75$ \mathrm{~km} / \mathrm{h} $, how long does it take the car to pass the train, and how far will the car have traveled in this time? What are the results if the car and train are instead traveling in opposite directions?

  An Olympic long jumper is5y m c9l-s5hxsk/ja l1 capable of jumping 8.0 m . Assuming his horizontal speed is 9.1 9/cl5k1mxlsayjh 5-s $\mathrm{~m} / \mathrm{s} $ as he leaves the ground, how long is he in the air and how high does he go? Assume that he lands standing upright - that is, the same way he left the ground. t =    y =   

  Apollo astronauts took a “nine iron”8oyin;0lqay ot4 ( ul; to the Moon and hit a golf ball about 180 m! Assuming that the swing, launch angle, and so on, were the same as on Earth where the same astronaut could hit it only 35 m, estimate the acceleration due to gravity onqy o 4n a;8tuliylo;0( the surface of the Moon. (Neglect air resistance in both cases, but on the Moon there is none!) g$_{moon}$  

  When Babe Ruth hit a ho x1waj+ iir7(vv9kgc:;hak 1ymer over the 7.5$-\mathrm{m}$ -high right-field fence 95 m from home plate, roughly what was the minimum speed of the ball when it left the bat?    Assume the ball was hit 1.0 m above the ground and its path initially made a 38$^{\circ}$ angle with the ground.

  The cliff divers of Acapulco push off horizontally from rock pla 5+y. zau0svci sux (,zk1a2pytforms about 35 m above the water, but they must clear rocky outcrops at water level that extend out into the water 5.0 m from the base of the cliff directly under their launch point. See Fig. below. What minimum pushoff speed vs 2zu5y(i uk+ zsa1.y0,xap cis necessary to clear the rocks?    How long are they in the air?   

  At serve, a tennis player aims to hitj.dbc(h0 ahbj,1 mt 6o the ball horizontally. What minimum speed is required for the ball to clear a(6t0 h1bo.jd,cmbjh the 0.90-m-high net about 15.0 m from the server if the ball is “launched” from a height of 2.50 m? $\approx$    Where will the ball land if it just clears the net (and will it be “good” in the sense that it lands within 7.0 m of the net)? How long will it be in the air? $\approx$    See Fig. below.

  Spymaster Paul, flying ;; hkqof tw.uz5k7y:m a constant $215 \mathrm{~km} / \mathrm{h}$ horizontally in a low-flying helicopter, wants to drop secret documents into his contact's open car which is traveling 155$ \mathrm{~km} / \mathrm{h}$ on a level highway 78.0 m below. At what angle (to the horizontal) should the car be in his sights when the packet is released (Fig. below) ?



$\theta=\approx $   $^{\circ}$

The speed of a boat in still water is v . The boat is to make a round trip ine ye/:qe7cc(veoy/if ) f rp(( a river whose current travels at speed ur e(7:y )/ qcp (/foifeyee(vc . Derive a formula for the time needed to make a round trip of total distance D if the boat makes the round trip by moving (a) upstream and back downstream, (b) directly across the river and back. We must assume u

  A projectile is launched from ground level to the top of a clif3iay3d, w3xng2em 6 tqf which is 195 m away and 155 m hm q6 x,wgdaneiy 3t233igh (see
Fig. 3-47). If the projectile lands on top of the cliff 7.6 s after it is fired, find the initial velocity of the projectile (magnitude and direction). Neglect air resistance.



$v_{0}$=    $ \mathrm{~m} / \mathrm{s}$
$\theta$=    $^{\circ}$

  

(a) A skier is accelerating down a 30.0$^{\circ}$ hill at 1.80$ \mathrm{~m} / \mathrm{s}^{2} $ (Fig. 3-48). What is the vertical component of her acceleration?    $m/s^2$
(b) How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation change is 335 m ?    $s$

  A basketball leaves a player's hands at a height o. hl00bbtmwdb)t -6myf 2.10 m above the floor. The basket isb.)b 06-bdt h0mt mlwy 2.60 m above the floor. The player likes to shoot the ball at a 38.0$^{\circ}$ angle. If the shot is made from a horizontal distance of 11.00 m and must be accurate to $\pm 0.22 \mathrm{~m}$ (horizontally), what is the range of initial speeds allowed to make the basket?      

  A high diver leaves the end of a zh(e7avhk/r/i)my)st -4vo 2h(ecbd$5.0-\mathrm{m}$ -high diving board and strikes the water 1.3 s later, 3.0 m beyond the end of the board. Considering the diver as a particle, determine
( a ) her initial velocity, $ \overrightarrow{\mathbf{v}}_{0}$ ,    ${ }^{\circ} \text { above the horizontal }$
(b) the maximum height reached,   
(c) the velocity $ \overrightarrow{\mathbf{v}}_{\mathrm{f}}$ with which she enters the water.    $^{\circ} $ $\text { (below the horizontal) }$

  A stunt driver wants to make his car jump over eight cars parked sidu c q +*uw i9toy3/rn9r3,.gihg3b8 wba.xm xje by side below a wg qmx9uaj+3uyg8* 9n 3,/ .wcrtii3xhrbo.bhorizontal ramp

(a) With what minimum speed must he drive off the horizontal ramp? The vertical height of the ramp is 1.5 m above the cars, and the horizontal distance he must clear is 20 m .    $m/s$
(b) If the ramp is now tilted upward, so that "takeoff angle" is $10^{\circ}$ above the horizontal, what is the new minimum speed?    $m/s$

  A batter hits a fly ball which leaves tpqb7(qh hm a)9he bat 0.90 m above the ground at an angle of qh(7)qp9 mah b $61^{\circ}$ with an initial speed of $28 \mathrm{~m} / \mathrm{s} $ heading toward centerfield. Ignore air resistance.
(a) How far from home plate would the ball land if not caught?   
(b) The ball is caught by the centerfielder who, starting at a distance of 105 m from home plate, runs straight toward home plate at a constant speed and makes the catch at ground level.

Find his speed.   

  At t=0 a batter hits a l0,mwsxl8ytmv+jt +28 l-enat 9ofb4 baseball with an initial speed of $32 \mathrm{~m} / \mathrm{s}$ at a $55^{\circ}$ angle to the horizontal. An outfielder is 85 m from the batter at t=0 , and, as seen from home plate, the line of sight to the outfielder makes a horizontal angle of $22^{\circ}$ with the plane in which the ball moves (see Fig. 3-50). What speed and direction must the fielder take in order to catch the ball at the same height from which it was struck? Give angle with respect to the outfielder's line of sight to home plate. [Hint: the law of sines may help]


v =    $m/s$
angle =    $^{\circ}$

  A ball is shot from the top of a building with an initial velocity of 18snj *8m8+n9g pauk vu2$ \mathrm{~m} / \mathrm{s}$ at an angle $\theta=42^{\circ}$ above the horizontal.
(a) What are the x and y components of the initial velocity? $v_x$ =    $v_y$ =   
(b) If a nearby building is the same height and 55 m away, how far below the top of the building will the ball strike the nearby building?   

  You buy a plastic dart gun, and being a clever physics student you decide rv5ofkkt0qb4 tevf+r*vn+ *2to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 4.0 s for the dart to lanb42vrf* rtvf+vq nk05toke+* d back at the barrel. What is the maximum horizontal range of your gun?