What is the evidence that sound travelso bxx.nex)68e as a wave?

What is the evidence that sound is a nq7 uii:r0z+kv l0 9vfxi9cc-form of energy?

Children sometimes play with a h7fml3 ol gt6+mbk: x50f 2wvidv-1ofhomemade “telephone” by attaching a string to the bottoms of two paper kdw2b x-f6 3ogv 75llm+10 f :vomfhitcups. When the string is stretched and a child speaks into one cup, the sound can be heard at the other cup (Fig. 12–29). Explain clearly how the sound wave travels from one cup to the other.

When a sound wave passes from air into water, do you expe . hx1ky9z/ fptndi6x1r79g tqct the frequency or wavelength ttyxq/1 .g91izdh k n9f6rp7xto change?

What evidence can you give that the spe;xtsavf d 9l8;pm.id*ed of sound in air does not depend significantly op; dis8al.x;d9*t f mvn frequency?

The voice of a person who has inhalej;2oc,xoh;*crl+z xd by 62e cd helium sounds very high-pitched. Why?

How will the air temperature in a room affect tcv-r/g/ tyf8 9rb 6lht-rw5tqhe pitch of organ pipes?

Explain how a tube might be used as a filter to reduce the amplitude of sjn1q *clwx5 q0u6wi .mounds in various frequency ranges. (An example is a car mu* wuq nm0.1x56jliqwcffler.)

Why are the frets on a guitar (Fig. 12–30) spaced closerm5nuid/ 6rzumw)y8uq8/qh y 9e ; t9td together as you move up the fingerb)y8dh; 5muz96y/er8 un iwtd uq/9tqmoard toward the bridge?

A noisy truck approaches you from behind a building. Initially you heac :n,m .g/y5 kky-w:xqbn,ottgn9 4dor it but cannot see it. When it emerges and you do see it, its sound is suddenly “brighter” — you hear mornc,ngty/4:w ,onokgyxt q9 .b md5-:ke of the high-frequency noise. Explain. [Hint: See Section 11–15 on diffraction.]

Standing waves can be said to be due to sn1 ley4stsmdh( k*gy1i;)6- ad/ b cd“interference in space,” whereas beats can be said to be due to “interference in btsg d;idnl s ym*d/ys(c14e16- a h)ktime.” Explain.

In Fig. 12–16, if the frequency of the speakers were lowered, would the p;u7hwrtr 8w1t50,ssos4p 02ht mqv v uoints D and C (where desrwtvhqw u0 pstr 52h04so8u 7t m;1v,structive and constructive interference occur) move farther apart or closer together?

Traditional methods of protecting the hearing of peyrx w.*wvu yhs3xt(.j( b 8w3rople who work in areas with very high noise levels have consisted mainly of efforts to block or reduce noise levels. With a relatively new technology, headphone r..wybht8w3 u xsvj3(r(*wy xs are worn that do not block the ambient noise. Instead, a device is used which detects the noise, inverts it electronically, then feeds it to the headphones in addition to the ambient noise. How could adding more noise reduce the sound levels reaching the ears?

Consider the two waves shown in Fig. 12–31y 2 nibq2zhw(lsd4 (2f. Each wave can be thought of as a superposition of two sound waves with slightly different frequencies, as in Fig. 12–18. In which of the waves, (a) or (b), are the two com 2di b42h nwfs(2qly(zponent frequencies farther apart? Explain.
(12-31)
(12-18)

Is there a Doppler shift if the source and observer move in thgh(vzvt, a1l1:rl cf8e same direction, with the same ve:c( h1lft 18arvlv,z glocity? Explain.

If a wind is blowing, will lko .t z2x*.xgx c4pkc; w30fcthis alter the frequency of the sound heard by a person at rest with respect to the source? Is the wavelength or velocity chap xg l4 z .kxcwc*xct32;.f0oknged?

Figure 12–32 shows various pozj/0tvvh* bpsp453zbsmr2o ; sitions of a child in motion on a swing. A monitor is blowing a whistle in front of the child on the ground.;/2 pjmt5v sb4ozzvs3h bp r0* At which position, A through E, will the child hear the highest frequency for the sound of the whistle? Explain your reasoning.

  A hiker determines the length of a lake by listening for the echo o *i,2erreqfa0b1 )scnf her shout reflected by a cliff at the far end of the lake. Sn2,bf e0a1rsice *)qrhe hears the echo 2.0 s after shouting. Estimate the length of the lake. $\approx$    $ \times10^{ 2}$m

  A sailor strikes the st2 vz2nn; gjx/ide of his ship just below the waterline. He hears the echo of the sound reflected from the ocean floor directly below 2.5 s later. How deep is the ocean at this point? Assume the speed of sound in seawater is 1560 m/s (Table 12–1) and does not vary signjtz ;nxv2g/n 2ificantly with depth.    $ \times10^{ 3}$m

  (a) Calculate the waverifw,:6v-j7b n0lt n wlengths in air at 20 $^{\circ} $ C for sounds in the maximum range of human hearing, 20 Hz to 20,000 Hz. $\lambda_{20Hz}$ =    m $\lambda_{20kHz}$ =    $ \times10^{-2 }$ m(b) What is the wavelength of a 10-MHz ultrasonic wave?    $ \times10^{ -5}$m

  An ocean fishing boat i/kx8kk*bfc:rcdf y :;s drifting just above a school of tuna on a foggy day. Without warning, an engine bacb:/kf*:; d8k fc yrkcxkfire occurs on another boat 1.0 km away (Fig. 12–33). How much time elapses before the backfire is heard (a) by the fish,    s (b) by the fishermen?    s

  A stone is dropped from the to3c eq n**. ccs;alf w:eu/wzo4p of a cliff. The splash it makes when striking the water below is heard 3.5 s later.e/zwfw 3u.;ceqa oc4*:csl* n How high is the cliff?    m

  A person, with his ear to the ground, sees a huge stone strike the dtamb*) 1piss: 3wth3du.;t bj4 asp8 concrete pavement. A moment later two sounds are heard from the impact: one travels in the air and the other in the concrete, and they are 1.1 s apart. How far away did the impact ocpuj.spttab)3 hd mi tbs ; d4s:a*18w3cur? See Table 12–1.$\approx$    $ \times10^{2 }$m

  Calculate the percent erj72v6m /hwqs lror made over one mile of distance by the “5-second rule” for estimating the distance from a lightning s/slm67 hwqjv2 trike if the temperature is (a) 30 $^{\circ} $ C, $\approx$    % (b) 10 $^{\circ} $ C$\approx$    %.

  What is the intensity of a sound at the pain leve4d aifc1fy,i( l of 120 dB?    W/$m^2$Compare it to that of a whisper at 20 dB.    $ \times10^{-10 }$W/$m^2$

  What is the sound leve sjn)sut/o4 ux0 7ka.al of a sound whose intensity is $2 \times10^{ -6}W/m^2$?    dB

  If two firecrackers produce a sound level of 95 dB when fired simultannix6qhg ad nwmznae-x5ly;p 2. 7 (b/0eously at a certain place, what will be the sound level if only one is exploded? [Hint: Add intensities, not dB’6;ybx/n 5ze.nh0qmaainxg7(d- 2lwp s.]    dB

  A person standing a cerir;;s5q5 zx3 z-9d tby).h nte/jxwdz tain distance from an airplane with four equally noisy jet engines is experiencing a sound level bordering on pain, 120 dB. What sound level would this person experience if the captain shut down all buxn3; ) 59zhd rqdtzwtsj/ yzbex;.i -5t one engine? [Hint: Add intensities, not dB’s.]    dB

  A cassette player is said to 7uuak,3 ky3wwkx1zf epak.-we+3yy2have a signal-to-noise ratio of 58 dB, whereas for a CD player it is 95 dB. What is the ratio of intensities of the signal and the background noise f+ k321 p ae.wak- ,yy33kuzfxwkuwye7or each device? 58 dB =    $ \times10^{ 5}$ dB =    $ \times10^{9 }$

  (a) Estimate the power output of sound from a person s+g ytlu gee05w5,7c kspeaking in normal conversation. Use Table 12–2. Assume the sound spreads roughly uniformly over a sphere centered on +tkl5,0w5gy ceesu7g the mouth. $\approx$    $ \times10^{ -6}$W(Round to the nearest integer)
(b) How many people would it take to produce a total sound output of 100 W of ordinary conversation? [Hint: Add intensities, not dB’s.]$\approx$    $ \times10^{7}$people

  A 50-dB sound wave strikes an eardrum who3nf8xg dxl rs6n/ylwbiwy, m 0x2/* f/se area is $5 \times10^{ -5} m^2$ (a) How much energy is absorbed by the eardrum per second?    $ \times10^{-12}$W (b) At this rate, how long would it take your eardrum to receive a total energy of 1.0 J?   $ \times10^{ 3}$yr

  Expensive amplifier A is rated at 250 W, while the morh 0tcqiuz51xql0 p-ktfo/w9qlaa :2qr 2k88ie modest amplifier B is rated at 40 W. (a) Estimate the sound level in decibels you would expect at a point 3.5 m from a loudspeaker connected in turnw0lp5a fhq- qoli1a i2tk:c/xkr 8qu09 z8q2t to each amp.$I_{250}$ =    dB $I_{40}$ =    dB (b) Will the expensive amp sound twice as loud as the cheaper one?

  At a rock concert, a dB meter registered 130 d64vl /r)mzyu vB when placed 2.8 m in front of a loudspeaker on the stage. yz4 )lr/muvv6
(a) What was the power output of the speaker, assuming uniform spherical spreading of the sound and neglecting absorption in the air?$\approx$    $ \times10^{2 }$W(“Round to one decimal place)
(b) How far away would the sound level be a somewhat reasonable 90 dB?   $ \times10^{2}$m

  Human beings can typically detect a diff4 y ppx m2*gpom ,t.rin-50mhwerence in sound level of 2.0 dB. What is the ratio of the amplitudes of two sounds whose 2n0mm,rm ip-p.* oh5gtxp wy 4levels differ by this amount? [Hint: See Section 11–9.]$\approx$   

  If the amplitude of a sound wave is tripled, (a) blb )vd 5: 09oj-wqnm;/.nc pmry9zh7-ds iuuuy what factor will the intensity u-) lvs9p;:-yzbu /uo7i0nm 9 . jmcqrdnd h5wincrease? Increase by a factor of    (b) By how many dB will the sound level increase?    dB

  Two sound waves have equal displacemex bs fo9teww(,a9 v:r(nt amplitudes, but one has twice the frequency of the other. What is the ratio of their inr(t(9,bave o w:s9fx wtensities?   

  What would be the sound level (in dB) of a soune nx1i3rm a kpktn 1vt70ve6o;)l ,2hnd wave in air that corresponds to a displacement amplitude of vibrating air mole6 0a1v nm;k1ntl hepio )2x73nv ,erktcules of 0.13 mm at 300 Hz?    dB

  A 6000-Hz tone must have what s 8p kgk/(ew54usjusa. 7l.glu 8)kt erqac)-d ound level to seem as loud as a 100-Hz tone that has a 50-dB sound level? (See Fig.s.7.kauw8ap(e45us gk/e-j dl) 8tlrqug )c k 12–6.)    dB

What are the lowest and highest frequencies that an ear can detect when the sou l6tgxbp0 .z :b2 xg5-d+uujfand level is 30b ub0zpglxf+u.2 -6ja 5: gtdx dB? (See Fig. 12–6.)

  Your auditory system can accom *wunmh+4/t lgmodate a huge range of sound levels. What is the ratio of highest to lowest intensity at hwt+ml gu*n4/
(a) 100 Hz,$ \times10^{ a }$ a =   
(b) 5000 Hz? $ \times10^{ a }$ a =   
(See Fig. 12–6.)

  The A string on a violin has 5m*k i2ttu vx(v;4l)owvry(o a fundamental frequency of 440 Hz. The length of the vibrating portion is 32 cm, and it has a xur yo;(v v vm)*4otlk2t(5wimass of 0.35 g. Under what tension must the string be placed?    N

  An organ pipe is 112 c3czeb;2fuhqj) 7c9 4a0n slk vm long. What are the fundamental and first three audible overtones if the pipe 0;zslb3k49 ejah f 7q2vucc) nis
(a) closed at one end, $f_{1}$ =    Hz $f_{3}$ =    Hz $f_{5}$ =    Hz $f_{7}$ =    Hz
(b) open at both ends?$f_{1}$ =    Hz $f_{2}$ =    Hz $f_{3}$ =    Hz $f_{4}$ =    Hz

  (a) What resonant freq0t -wj lq5/7tne-9 8zicbzp czuency would you expect from blowing across the top of an empty soda bottle that is 18 cm deep, cp9-t8/-bq ewl z jtz0ic5nz7if you assumed it was a closed tube?    Hz (b) How would that change if it was one-third full of soda?   Hz

  If you were to build a pipe orgw kdo5cd,p+);vz7 njs:cf g n2an with open-tube pipes spanning the range of human hedzs5n +27dpg;own,ccfv jk: )aring (20 Hz to 20 kHz), what would be the range of the lengths of pipes required? $L_{20 Hz}$ =    m $L_{20 kHz}$ =    $ \times10^{ -3}$m

  A tight guitar string has a frequency of 5vqa/yp(qg2w ogw7s;*40 Hz as its third harmonic. What will be its fundamental frequency if it is fingered at a length of only wgsyqq 7o;*gp (w a2v/60% of its original length?   Hz

  An unfingered guitar +)0b yoye kj t)efk kn6yrd/3nqp3+;pstring is 0.73 m long and is tuned to play E above middle C (330 H/) 6ek3;)okyq +bydkptpjy3frnne+0 z).
(a) How far from the end of this string must the finger be placed to play A above middle C (440 Hz)?$\approx$    m
(b) What is the wavelength on the string of this 440-Hz wave? $\approx$    m
(c) What are the frequency and wavelength of the sound wave produced in air at 20°C by this fingered string?    Hz    m

  (a) Determine the length of an open organ pipe that emits midkybyl7q4d0 * abrwyu-e4d *4 udle C (262 Hz) when the temperature qryewlu7ad*d4*-40kbyy4u b is 21 $^{\circ} $ C.    m
(b) What are the wavelength and frequency of the fundamental standing wave in the tube?    Hz    m
(c) What are $\lambda$ and $f$in the traveling sound wave produced in the outside air?    Hz    m

  An organ is in tune r b:0 )typz,7a 1 ;xzlzbx1eqpat 20 $^{\circ} $ C. By what percent will the frequency be off at 5.0 $^{\circ} $ C?    %

  How far from the mouthpiece of the flute in Example 12–10 should the hole 4bk kc ;x:xfu8be that must be uncovered to play D above middle C akbkuc;: 8xx4ft 294 Hz?    m

  (a) At T = 20 $^{\circ} $C how long must an open organ pipe be to have a fundamental frequency of 294 Hz?    m
(b) If this pipe is filled with helium, what is its fundamental frequency?   Hz

  A particular organ pipe can reson2o19xbd.z gxa ate at 264 Hz, 440 Hz, and 616 Hz, but not at adx2x g.1zboa9 ny other frequencies in between. (a) Show why this is an open or a closed pipe. ----待选择----closedopen  (b) What is the fundamental frequency of this pipe?   Hz

  A uniform narrow tube 1.80 m long is open at both ends. cn)u b4 )du gbd3mrw k+,lw(iz;(a8keIt resonates at two successive harmonics of frequenl)ciad34d wzbe uw,b(n)k mku;8 +gr(cies 275 Hz and 330 Hz. What is
(a) the fundamental frequency,    Hz
(b) the speed of sound in the gas in the tube?$\approx$    $ \times10^{2 }$ m/s

  A pipe in air at 20 subbiy25 t 2a .8asn)t$^{\circ} $ C is to be designed to produce two successive harmonics at 240 Hz and 280 Hz. How long must the pipe be, and is it open or closed?  ----待选择----closedopen  f =    m

  How many overtones are present n0j -4vog.tu zwithin the audible range for a 2.14-m-long organ u .jzn gv-0o4tpipe at 20 $^{\circ} $ C
(a) if it is open,   
(b) if it is closed?   

  The human ear canal is a53cpl9z 94j,.cj o hpkspb q5ypproximately 2.5 cm long. It is open to the outside and is closed at the other end by the eardrum. Estimate the frequencies (in the audible range) of the standing waves in the ear canal. What is the relationship of your answer to the inform3z,jljp y5qk4ps. 95o pbhcc 9ation in the graph of Fig. 12–6? $f_1$ =    Hz $f_3$ =    Hz $f_5$ =    Hz

  A piano tuner hears one beat ev.; ivok pdc4rp*g.3+c9o m uusery 2.0 s when trying to adjust two strings, one of which is soundp4g;3o9 dskc.mip cu*ru +v.o ing 440 Hz. How far off in frequency is the other string? ±    Hz

  What is the beat frequency if middle r a), v,wydk9a0lldu (C (262 Hz) and $C^＃$ (277 Hz) are played together?    Hz What if each is played two octaves lower (each frequency reduced by a factor of 4)? $\approx$    Hz

  A certain dog whistle operates at 23.5 kHz, whiln i m1 *sz2q+xws7pxk5e another (brand X) operates at an unknown frequency. If neither whistle can be hxs*+1 sn pm7wx5k zqi2eard by humans when played separately, but a shrill whine of frequency 5000 Hz occurs when they are played simultaneously, estimate the operating frequency of brand X.    kHz

  A guitar string produces 4 beats/s when sounded with a 350-Hz tuning fork asta,zydw ze,8;f jc qyd5y955nd 9 beats/s when sounded with a 355-Hz tuning fork. What d;5zyytzc,j s5ed,wq 98 yf5a is the vibrational frequency of the string? Explain your reasoning.    Hz

  Two violin strings are tuned to the same frequency, 294 Hz. The teny7 jyyhl0w 1/ksion in one string is then decreased by 2.0%. What will be the beat frequency yj/y w0y1kh7lheard when the two strings are played together? [Hint: Recall Eq. 11–13.]    Hz

  How many beats will benbyd*gxil61j01 c 8 akl7* psi heard if two identical flutes each try to play middle C (262 Hz), 1 ikbjp *xl1ng7*iy06 sdc8albut one is at 5.0°C and the other at 25.0 $^{\circ} $ C?    beats/sec

  You have three tuning forks, A, B, and C. Fork B has a frequenclow*wcgc2.0pq:c4gu y of 441 Hz; when A and B are sounded tc0qup gwcwg. o4c*:l2ogether, a beat frequency of 3 Hz is heard. When B and C are sounded together, the beat frequency is 4 Hz. What are the possible frequencies of A and C? What beat frequencies are possible when A and C are sounded together?$f_A$ =    Hz or    Hz
$f_C$ =    Hz or    Hz
|$f_A$-$f_C$|    Hz or    Hz

  Two loudspeakers are 1.80 m apart. u355+pmaktdk A person stands 3.00 m from one speaker and 3.50 m from the odka5 u+mk3p t5ther.
(a) What is the lowest frequency at which destructive interference will occur at this point? f =    Hz
(b) Calculate two other frequencies that also result in destructive interference at this point (give the next two highest). Let T = 20 $^{\circ} $C $\approx$    Hz $\approx$    Hz

  Two piano strings are suppz37ru kx+d1l ( yrj0eoosed to be vibrating at 132 Hz, but a piano tuner hears three beats every 2.0 s when they are played together. (a) If one is vibrating at 132 Hz, what must be the frequency z0(3lr+ od 1rkxuye7jof the other ?   Hz    Hz
(b) By how much (in percent) must the tension be increased or decreased to bring them in tune?   % increase    %decrease

  A source emits sound of wavelengths 2.64 m and 2.7610fd+es: ; 5qsgrmkgd m in air. How many beats per second will be heard? (Assume T :dsk0s;1 q+gr m5g fde= 20 $^{\circ} $C)$\approx$    Hz(round to one decimal place)

  The predominant frequency of a certain fire engine’sssg+ 1y )x0qt 0pyjm;n siren is 1550 Hz when at rest. What frequency do you detect ify ;1 g0njpxst+qy ms)0 you move with a speed of 30 m/s
(a) toward the fire engine,    Hz
(b) away from it?   Hz

  You are standing still. What frequency do you detect if a htz w 0cbg5u00gn *m3z*c q6klfire engine whose siren emits at 1550 Hzc50w cl60*ut0bkzn* m h 3zggq moves at a speed of 32 m/s
(a) toward you,    Hz
(b) away from you?   Hz

  (a) Compare the shift in frequency if a 2000-f.mw+9vav8qds +3 qrb 0 ax,gbHz source is moving toward you at 15 m/s versus you moving toward it at 15 m/s Are the two frequencies exactlyaqwv ., 8abg xb+9dfs0q+r3vm the same? Are they close? $f'_{source\,movine}$    Hz
$f'_{observe\,movine}$    Hz
(b) Repeat the calculation for 150 m/s and then again
$f'_{source\,movine}$    $ \times10^{ 3}$Hz
$f'_{observer\,movine}$    $ \times10^{ 3}$Hz
(c) for 300 m/s What can you conclude about the asymmetry of the Doppler formulas?$f'_{source\,movine}$    $ \times10^{ 3}$Hz
$f'_{observer\,movine}$    $ \times10^{ 3}$Hz

  Two automobiles are eq1;-bm i*kwfhmuipped with the same single frequency horn. When one is at rest and the other is moving toward the first at 15 h*;b ki1 mw-fmm/s the driver at rest hears a beat frequency of 5.5 Hz. What is the frequency the horns emit? Assume T = 20 $^{\circ} $C    Hz

  A bat at rest sends out ultrasonic sound waves at 50.0 kh rq)bq d; d4dt,ak-y2Hz and receives them returned from an object moving directly away from it at 25 m/s What is the rec2k-br; t yqq4hddd,) aeived sound frequency?    $ \times10^{ 4}$ Hz

  A bat flies toward a wall at a ss(53 5ooqy / daar)uyyb( u2vlpeed of 5 m/s As it flies, the bat emits an ultrasonic sound wave with frequency 30.0 kHz. What frequency does the bat hear )(vy52 a/os u3ly(yd5uobq arin the reflected wave?    $ \times10^{4}$ Hz

  In one of the original Doppler experiments, a tuba was played on a moving flat td knk)cl44s eo2+ 0wowrain car at a frequency of 75 Hz, and a second identical tuba played the same tone while at rest in the rsc4oew l nw2o04 +dk)kailway station. What beat frequency was heard if the train car approached the station at a speed of 10 m/s ?   Hz

  A Doppler flow meter uses ultrasound waves to measure jp(3 9 u/zw viq 3wu3igt27egrblood-flow speeds. Suppose the device emits sound at 3.5 MHz, and the speed of sound in human tissue is taken to be 1540 m/s What is the expected beat frequency if blood is flowing in large leg 9tz gg2/ wqv 7p(u3rej3i i3uwarteries at 2 cm/s directly away from the sound source?   Hz

  The Doppler effect using ultrasonic w 6 0rhc88. frhaiflev3aves of frequency $2.25 \times10^{ 6}$Hz is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of 500 Hz is observed. Assuming that the speed of sound in tissue is $1.54 \times10^{3 }$ m/s calculate the maximum velocity of the surface of the beating heart.    m/s

  A factory whistle emits sound of frequency 570 Hz. When the wind velocity is 12ibumuwhq:6,pc 8y i,) 6fbg0 p m/s from the north, what frequency will observers hear who are located, at rbih6g6bip 0mc)q,8w :fu puy, est,
(a) due north,   Hz
(b) due south,   Hz
(c) due east,    Hz
(d) due west, of the whistle? What frequency is heard by a cyclist heading    Hz
(e) north    Hz
(f) west, toward the whistle at 15 m/s?   Hz
Assume T = 20 $^{\circ} $C

  How fast is an object movin.n8q)ze2 2 gxggfw96 i:vflvz;p f1bxg on land if its speed at 20 $^{\circ} $C is Mach 0.33?    m/s
(b) A high-flying jet cruising at 3000 km/h displays a Mach number of 3.2 on a screen. What is the speed of sound at that altitude?    m/s

  An airplane travels at Mach 2.3 where the speed of sound is u,w 7he;y7i)e60qfv gh yy u6o2s2zly 310 m/s (a) What is the angle the shock wave makes with the direction of the airplawyuf,yh2 lq7;0h zevgi27yy6 u) seo6ne’s motion?    $^{\circ} $
(b) If the plane is flying at a height of 7100 m, how long after it is directly overhead will a person on the ground hear the shock wave? $\approx$    s (Round to the nearest integer)

  A space probe enters the thin atmosphe-a4 dodq diml (u9-0 1kkv-fslre of a planet where the speed of sound is only about 35 msi a d(-k 091l-mkvdl4qdu o-f/s
(a) What is the probe’s Mach number if its initial speed is 15000 km/h$\approx$    (Round to the nearest integer)
(b) What is the angle of the shock wave relative to the direction of motion?    $^{\circ} $