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Determine the sp ed at which he was traveling off the ramp, the horizontal di tanee he trav I before striking the ground, and the maximum height h attain. It is also pos ible to find a without having to calculate 'r or aJ As shown in Fig. F lb tb. F Fl Since a. Fl F Consider particle A and B which move along the arbitrary paths shown in Fig.

Al 0, note th e horizontal di tance between ucce i e photo of the yellow balli constant since the eloci ty in the horizontal direction remain con tant.

To summarize, problem involving the motion of a projectile can have at most three unknown ince only three independent equation can be written' that is, one equation in the horizontal direction and two in the vertical direction. Between any two points on the path specify th given problem data and identify the three unknown. In all cases the acceleration of gravit act downward and equal 9. The particle s initial and final velocities hould b represent d in terms of their x and y components.

R member that po itive and n gati e pOSitIOn , velocity and acceleration component always act in accordance with their a ociated coordinate directions. Depending upon the known data and what is to be determined , a choice should be made as to which Hue of the following four equations should be applied between the two point on the path to obtain the most direct olution to the problem.

In the vertical or y direction only fwO of the following three equations can be used for , olution. Tn thi way th e location of the accumulated pile can be determined. Rectangular coordinate are u d for the analY 'i ince the acceleration is only in the crtical direction.

For example, if the particle S final elocity ' i not needed then the fir t and third of the e equation will not be u efu l. If the height of the ramp is 6 m from the floor determine the time needed for the sack to strike the floor and the range R where sacks begin to pile up. The origin of coordinates is established at the beginning of the path point A , Fig. H re we do not need to detemline VB'. Vertical Motion. TIle calculation for tAB also indicates that if a sack were released from rest at A it would take the sam amount of tim to trike the floor at C Fig.

If the tube is oriented at 30 from the horizontal d termine how high h, the chips strik the pile if at this instant they land on the pile 20 ft from the tube. During a race it was observed that the rider shown in Fig. Determine the sp ed at which he was traveling off the ramp, the horizontal di tanee he trav I before striking the ground, and the maximum height h attain.

As shown in Fig. In order to find the maximum height h we will con ider the path A , Fig. Here the three unknown are the time of flight tAC the horizontal distance from A to C and the height h. If the x component of the particle's ve locity i.. D e termin e th maximum he ight II it reache. A bal! If it i requir d to clear th e wal! D t rmin e th range R, and th e p ed wh n th ball strik. D e termin e th e peed at which th e ba k tbal! Water is prayed at an ang le of from the slope y.

I 0, prove that the path of the particle is e lliptical. If the particle i. Also, what i the x, y, Z coordinate po ition of the particle at this instant?

The car travel. Detenn in th e magnitude of th e displac m nt of th e car and th e di tance trav I d. A car tTavel east 2 km fOT 5 minute, then north 3 km for minutes, and then we t 4 km for 10 minute.

Determi ne the total di. AI 'o, what i ' the magnitude of the average velocity and the average peed? A car travel ing along the straight portions of the road ha. Deter min e its average speed when it goes from A to D. The po ition of a crate. If it take 3. If th e component of velocity along the x axis i. Th motorcycle tra Is with con ' tant p d Vo along th e path th at for a hort di tance, takes the form of a in e curve. D e te rmin e the x a nd y components of its velocity at any in.

Dur in g a race the dir t bike was ob ' rved to leap up off th e mall hill at A at an angle of 60 with th e hori zontal. If th e point of landing is 20 f t away, determine the approxim ate peed at which the bike wa tra elin g ju t before it left th gro und. It i ob erved that the time for th e ball to trike th ground at B i ' 2. D eterm ine the p ed VA and angle OA at. The girl always throw th toys at an angl of 30 fro m point A as shown. D e termine th e time be twee n throw 0 th a t both toy tr ike th e edge of the pool B and C at th e arne instant.

With what peed mu. De termin th di tance d to wh r it trike th e slope d ground. The acceleration du gra vity is g. A projectile is given a velocity vo. The acce leration due to gravity i g. De te rmin e th e time th e ball i in th e air and th e angle f of the kick. From a videotape, it was ob erved that a player kicked a football ft during a mea ured time of 3.

De termin the initial peed of the ball and th e angle f at which it wa ' kicked. D e termin e the mllllmUm height o n the wall to which the fir fi ght r can project water fro m the ho 'e, so th at th water trike th wa ll horizontall y.

Determ in the malle t angl O, mea ured above the horizontal, that the ho ' hould be direct d 0 th at th e water stream tr ikes th e bottom of the wall at B. Th e ba ll pa d through th e hoop e ven though it barely clear d the hands of the player B who attempted to block it. D tem,i ne th e di 'lance d to wh ere it will land. T he ball is th row n from the tower with a ve locity of 20 ftl a hown. Dete rmin e th e x and y coordi nate to wh re the ball tri ke th e lope. AI 0, d termi ne th e peed a t which the ball hit the ground.

If he trike th e gro un d at B. It i ob. If he trike th e gro und at B, det rm ine hi ' initial 'peed 'VA and th e p ed at which he trike the gro und. The projecti le i launched with a elocity vo. De termin the ra nge R , the rn a im um height h attain ed, pres' the re ults in term ' of the and th e tim of fli ght. The acceleration due to gravity i g. Detemline the horizontal ve locity ball at A 0 that it just clear. The man at It wi.

If each dart i thrown with a peed of 10 mi s, determine the angle Dc and fJo at which they. Determin th e point x, y where it trike the ground. Assume the ground has the shape of a parabola a shown. If it trik. A boy throw a ball at 0 in the air with a speed Vo at an angle fJ I.

Wh n th ball is directly overhead of player B he begins to run under it. Determine the can. The firema n wi he to direct th e flow of water from hi ho e to the fire at B. Determine two po. T he man tand 60 ft from the wall a nd throw. D etermine the a ngle at which he hould relea e th e ball 0 that it trik the wall al the highest poi nt po sib le.

What i this height? The room ha a ceiling he ight of 20 ft. When the path along which a particle travels i known then it i often convenient to describe the motion using nand t coordinate axe which act normal and tangent to the path, respectively, and at the in tant considered have their origin located at the particle. Planar Motion. Con ider the particle hown in ig. We will now consider a coordinate sy tern that ha its origin on the curve and at the instant considered this origin happ ns to coincide with the location of the particle.

The t axis is tangent to the curve at the point and is positive in the direction of increasing s. We will de1 ignat thi po iti e direction with the unit vect r U t. A unique choice for the normal axis can b made by noting that geometrically the curve icon tructed from a.

The normal axis n is perpendicular to the t axis with its positive sense directed toward the center of curvature 0 ', Fig. This positive direction which is always on the concave side of the curve will be designated by the unit vector u ll The plane which contains the n and t axes is ref rred to as the embracing or osculating plane, and in this case it i. Since the particle moves,S is a function of time. As indicated in Sec. As no ted above, the 0 culating plane is alway coincident with a plane curve; how ver.

To better under tand the e re ults consider the following two special ca e of motion.

Therefore, the normal component of acceleration represents the time rate of change in the direction of the veloCity. A a result of the e interpretations a particle moving along the cur ed path in Fig. As the boy swi ngs upward with a velocity v, hi motion can be analyzed u ing n-t coordinate. As he rises, the magnitude of hi velocity speed i decreasing.

If the particle mo e. A in the case of planar motion we will choose the positive n axis directed toward the path s center of curvature a'. This axis i referred to as the principal normal to the curve. With the nand t axes s defined, Eq. Remember though that U'I is always on the concave side of the curve.

The acceleration of the particle is the time rate of change of the velocity. TIlU , ' In order to determine the time derivative Ul note that a the particle move along the arc d in time dt, U I prcscrves its magnitude of unity; however, it direction changes, and become u;.

Substituting into Eq. TIlcse two mutually perpendicular components are shown in Fig. Provided the path of the particle i known. The po itive tangent axi acts in the direction of motion and the positi e normal axis i directed toward the path center of curvature. The particle s velocity is alwa s tangent to the path. The magnitude of elocity i found from the time derivative of the path function. Th relations between a, v, [ and s are the same as for rectilinear motion namely. Th normal component of acceleration is the result of the time rate of change in the direction of the velocity.

This component is alway directed toward the center of curvature of the path, i. The magnitude of this component is determined from. Motori t traveling along thi cloverlea f int erch ang e 'p eri nce a norm al acceleration due to the ch ange in direction of their ve locity.

Determine the direction of hi. Although the path ha been expres ed in terms of it x and y coordinates we can still e tabli h the origin of the n, taxes at the fixed point A on the path and determine the components of v and a along these axes Fig. By definition the velocity is always directed tangent to the path. Ther fore.

What i its peed at this in tant? TIlis coordinate ystem i selected since the path i known. Since all the velocity a a function of time mu t be determined fir t. Remember the velocity will alway be tangent to the path, whereas the acceleration v. If a box as in Fig. The po ition of the box at any in tant i d fined from the fixed point A u ing the po ition or path coordinate s, ig.

The acceleration is to be determined at Bothe origin of the n, t axe i at thi point. D e temlin e the dir ction of th e crate's velocity and the magnitude of the crate's accelera tion at thi in ta nt.

Fl F A ca r i trave ling a lon g a circul ar curve that has a radiu of 50 m. D e te rmin e th e m a ximum co n stant speed a race ca r ca n ha ve if the accelerati o n of the car ca nn o t exceed 7.

T he a uto mobi le is orig in a lly at re. D tem, ine th m agnitud of it. A car trave ls a lo ng a ho ri zo nta l c ircul ar curve d road tha t ha a ra diu of m.

D e te rmine it. D e term ine th e magnitude of th e acce le ratio n of th e roller coaste r at thi in ta nt a nd the di rectio n a ng le it ma ke. If it takes 45 s to make the turn, detemline the m agnitude of th e boat's acce leration duri ng th e turn. The car pas. If the car pa. A train is traveling with a con tan t speed of 14 m j along the cur ed path. D e termin e the magnitude of the acce lera ti on of th e front of the train, B, at.

D e termine the rat of increa e in th e train ' peed and the radius of curvature p of the path. If h tart from re 't at A , determin e the magnitudes of hi s ve locity and acceleration when he reache R.

If he tarts at V. A l 0, what i hi s initi al acc leration? Neglect th e ' ize of the car. Neglect the size of the car. T he j t pl ane trave l a lo ng the vertical paraboli c path. D e term ine the magnitude of accele rati o n of th e pla ne whe n it i. A boat i trave ling alo ng a ci rcul a r path ha ing a radius of 20 m. De te rmin e the magn itude of th e boat' acce lera tion whe n the. D etermin e th e magnitude of it acce lera tio n whe n it i, at poi nt J1. De te rmin th e magni tude of its acc leration whe n it is at point A.

Determine the point on the curve wher the maximum magnitude of acceleration Occurs and comput it ' value. The Ferris wheel turns uch that the. If th whe I tart from re t wh n: The two particles A and B tart at the origin 0 and travel in oppo. Determine the time when they collide and the magnitude of the acceleration of B just before thi happen. How far ha. D t rmine th magnitud of the particle" accel ration at thi in Lant. Determine the magnitude of the velocity and acce leration of the car when it r ache point B.

Solve for the velocity Vp and acceleration a p of the particle in term of their i, j , k component. Th e binorm a l is parallel to. The motorcycle travel along the e lliptica l track a t a con tant peed v. Jfmotion i re tricted to the plane then polar coordinate are u ed. We can speeify the location of the particle shown in Fig. TIle positive direction of the rand 8 coordinate are defined by th e unit vector u ,. At any instant the position of the particle defin d by the po ition ector. The instantaneous velocity v is obtained by taking the time derivative of r.

To evaluate Up notice that u, only changes it direction with respect to time, since by definition the magnitude of this vector is always one unit.

Hence during the time 6. The time change in U r is then 6. For small angles 6. Therefore, 6. The e components are hown graphically in Fig. The radial component Vr is a mea ure of the rate of increa e or decrease in the length of the radial coordinate i. To evaluate 0o, it is necessary only to find the change in the direction of Ue since its magnitude is always unity. For small angles this vector has a magnitude fluo "'" I flO and acts in th - U , direction' i.

Since a, and ae are alway perpendicular, the magnilllde of acceleration imp ly the po itive alue of. The direction is detennined from the vector addition of its two component. In general, a will not be tangent to the path , Fig. Cylindrical Coordinates. If the particle mo es a long a pace cur a hown in Fig.

The z coordinate i identical to that used for rectangular coordinate. Since the unit vector defining it direction u: Two type of problems generally occur:. U ing the chain rule of calculus we can then find the relation between rand 8, and between ',: Application of the ehain rule along with som examples, is explained in Appendix C. Th e piral motion of lhis gi rl can b follow ed by u ing cy lindrical co mp nent.

H re the r adial coo rclinat e r i constant , the tran ve r se coordinate J will increa e with tim e a the girl rotat about thc vcrt ical. Polar eoordinates are a suitable choice for olving problems when data regarding the angular motion of the radial coordinate r is gi en to de crib the particle' motion. A I 0 , orne path of motion can conveniently be de cribed in term of these coordinate.

To use polar coordinates, the origin is estab lished at a fixed point and the radial line ,. The tran er e coordinate 8 i mea ur d from a fixed reference line to the radia l line. Onc ,. See Appendix C. Motion in three dimen ion requi re. Since the angular motion of the arm is reported , polar coordinat s are chosen for the solution Fig. Here J is not related to 1', since the radius is constant for all o.

Velocity and Acceleration. It is fir t nece sary to specify the first and second time derivatives of rand o. Since r is constanl we have r. The e result are hown in Fig. The n, t axes are also shown in Fig. At the arne time, the collar B i Iiding outward along OA. The velocity is tangent to the path ; however the acceleration is directed within the curvature of the path , as expected.

Determine th e magnitud of the velocity and accel ratio n at which the pot appear to tra el acro. Polar coordinate will be u ed to.

To find the neces ary time derivatives it is fir t necessary to relate,. It is also pos ible to find a without having to calculate 'r or aJ As shown in Fig. This path is most unusual , and mathematically it is best expressed usi ng polar coordinates a done here, rathe r than must be determined rectangular coordinate.

Al 0 since 0 and then r, 8 c ordinates are an bvious choice. Vector a and vare hown in Fig. The car ha. Determine the angular velocily iJ of th radial lin OA at thi in Ian!. Detemline the radial and tran verse component of the peg's acceleration at thi in tant. Ft F A ball roll outward along the radia l groove 0 that it po ition i ,. F F Determine the magnitude of the velocity of the collars at thi. Peg P i. Determine the magnitude of the peg acceleration at thi s in. D termine the angular velocity at which the camera must turn in order to fo ll ow the motion.

A particle is mo ing a long a circular path having a radius of 4 in. Determine the particle s radial and tran vcr. D termine th e magnitude of locity and acceleration of th e car at thi ' in tant. An airplane is flyin g in a. A car i trav ling along th e circu lar curve of radius,. Determin e the magnitude of th car' e locity a nd ace I ration at this in rant. If a particle's po ition i. Determine th e radial and tran sver e component of th e car'.

Th e car travel along th circul ar cur e of radiu. D e termin e the angular rate of rotation j of the radial line r and the magnitude of the car's accelerat ion. The tim e rate of change of acc leration is referred to a th e jerk, whi ch i ' often u 'ed a a mea n of m a urin g pa engel' disco mfor t. Calculate thi vector, il, in term of its cylindrical components, u ing q. If it maintain a con 'ra nt. A particle i movin g alon g a circular path having a radiu of 6 in.

Th e particle tarts from re t at. Th e lotted link is pinned at O. D e te rmin e th velocity and acceleration of the particle a t th e in tant it leaves the slot in th e link i.

Determine the angular velocity of the camera tracking the car at thi. Wh e n it is at th e midpoint, its p e d i mlll j and it acceleration is 10 mm j 2 , Expr s the velocity and acceleration of th e washer at thi point in terms of it cylind rical component. A block moves outward along the lot in the platform with a p d of ,: If th angul ar velocity icon ta nt at 6, de temline the radi al and transver e components of velocity and acce leration of the pin. D etermin e the magnitudes of th e ve locity and accelera ti o n of th e rod A Bat thi.

Wh en f: D etenn ine the magnitude of the ve locity a nd acce leratio n of th e peg Pat thi. D etermine th e magnitude of it acce le ration. Th e ramp de cends a vertica l distance of 1 m for every full revolution.

In , wh ere t i in econd. D etermine the magni tude of the velocity and accelera tio n of the box at th e in. If it maintains a con.

For a short distance the train travel a lon g a track havin g the. If the a ngula r rate i. J Prob. Sketch the curve and show th co mponent o n the curve. In ome types of problem the motion of one particle will depend on the corresponding motion of another particle. Thi dependency commonly occurs if the particles, here represented by blocks are interconnected by inextensible cord which are wrapped around pulleys. For example the movement of block A downward along the inclined plane in Fig.

We can how thi mathematically by fir. Note that each of th coordinat aXel i 1 mea ur d from a fixed point 0 or fixed datum line, 2 measured along each inclined plane in the direction of motion of each block, and 3 ha a po iti e en e from the fixed datum to A and to B.

If the total cord length is iT, the two position coordinates are related by the equation SA. The negative sign indicates that when block A has a velocity downward i. B moves in the negative SB direction. In a similar mann r time differentiation of the vel cities yields the relation between the accelerations i. A more complicat d example is shown in Fig.

In this case, th position of block A is specified by SA, and the position of the end of the cord from which block B is su pended i defined by SB. As above we have chosen position coordinates which 1 have their origin at fixed points or datums 2 are measured in the directi n f moti n of each block, and 3 from the fixed datums are positive to the right for SA and po itive downward for B.

During the motion , the length of the red color d egment of the cord in Fig. U - 37a remains constant. This example can also be worked by defining the position of block B from the center of the bottom pulley a fixed point , Fig. The above method of relating the dependent motion of one particle to that of another can be performed using algebraic sealars or po ition coordinate provided eaeh particle moves along a rectilinear path.

When this i the ease only the magnitudes of the velocity and acceleration of the particles will change, not their line of direction. Position-Coordinate Equation.

Establish each position c ordinate with an origin located at a fixed pint r datum. It i not neces ary that the origin be the same for each of the coordinate ; however, it is important that each coordinate axis seleeted be directed along the path of motion of the particl.

Using geom try or trigonometry relat the position coordinates to the t tal length f the cord IT or to that portion of cord I which excludes the segments that do not change length as the particle mo e - uch a arc egmen wrapped over pulley. If a problem involves a syscem of two or more cords wrapped. Separate equations are written for a fix d I ngth of each cord of the y. Time Derivatives.

Th e motion of th e lirt on th is crane depend upon the motion of th e cable connect ed to th e winch which operates it. Two ucc ive time derivative of th po ition-coordinat equations yi Id the requir d velocity and acceleration equation which relate the motion of thc particle.

The signs of the tenns in these equations will be con istent with those that specify the positive and n gative sense of the position coordinates. There is one cord in this system having segments which change length. Position coordinates SA and S8 will be used since each is measured from a fixed point C or D and ext nd along each block' path of motion.

Tn particular, B i. The red colored egments of the cord in ig. TIle remaining length of cord I, i also con tant and is related to th changing position coordinates SA and: As sh wn the positions of blocks A and B are defined using coordinates SA and S8 ' Since the system has two cords with egments that change length it will be necessary t use a third coordinate, sc, in order to relate A to B. Tn other word , the length of one of the cord can be expre.

For the remaining cord lengths say II and 12 we have. Eliminating Vc produces the relationship between the motion of each cylinder. The position of point A is defined by SA and the position of block B is specified by SB since point E on the pulley will have the same mOlion as the block. Both coordinates are m a ured from a horizontal datum pa ing through the rued pin at pulley D. In tead by establishing a third position coordinate, sc, we can now expre s the length of one of the cord in term of 8 and sc.

Determine the velocity and acceleration of the afe when it reache the el ation of 10m. The rope is 30 m long and pa se over a mall pulley at D. However the end of the rope, which define the po itions of C and A are specified by mean of the x and y coordinate ince they must be measured from a fixed point and directed along the pachs of motion of the ends of the rope. Henc from q. Th constant velocity at A causes the other end of the rop to have an acceleration since VA cau e egment DA to change it.

Throughout thi chapter the ab olute motion of a particle has been determined using a ingJe fixed reference frame. There are many case , however where the path of motion for a particle is complicated so that it may be easier to analyze the motion in parts by u ing two or more frames of reference.

For example the motion of a particle located at the tip of an airplane propeller, while the plane is in flight is more easily de cribed if on ob erve fir. In this ection fran lating frames of reference will be considered for the analy i.

Consider particle A and B which move along the arbitrary paths shown in Fig. The absolute position of each particle r A and rB , is measured fr m the common origin 0 of the rued x y Z reference frame.

The origin of a second frame of reference x', y' z' i attached to and move with particle A. The axe. An equation that relates the velocitie of the particles is determined by taking the time derivative of the above equation' i.

It is important to not that since the x ' , y' 7 ' axes translate, th components of rBIA will not change direction and therefore the time derivative of these component will only have to account for the change in their magnitude. The time derivative of q. Here aBIA i the acceleration of B as seen by the observer located at A and translating with the x' y', Z' reference frame. When applying the relative velocity and acceleration equations it i first necessary to specify the particle A that i the origin for th translating x' y' Z' axes.

Usually this point has a known velocity or acceleration. These unknowns can be solved for either graphically, using trigonometry law of sine law of cosines or by r solving each of the three vector into rectangular or arte ian component thereby aenerating a.

The pi lot of the e jet planes fl ying clo 'e to one another mu t be aware of th eir relative po i tio n and velocitie at al l time in ord r to a aid a colli ion. This figure anticipates the answer and can be used to check it. We will assume these compon nts act in the positive x and y dir ctions. Plane A in Fig. D etermine the velocity and acceleration of B a m a ured by the pilot of A.

The origin of the x and y axe are located at an arbitrary fixed point.

Since the motion relative to plane A is to be determined th e translating frame of reference x ' , i attached to it, ig. Applying the relative-velocity equation in scalar form since the velocity vectors of both planes are parallel at the instant shown we have I. The vector addition is shown in Fig.

Plane B ha both tangential and normal components of acceleration since it i flying along a curved path. The so lution to this problem was possible using a translating frame of reference sinee the pilot in plane A is ' tran lating. The analysis for thi ca e is given in xamplc Also at this instant A has a decrea e in peed of 2 ml 2 and B ha an increa.

Determine the v locity and acceleration of B with re pect to A. U ing a Carte ian vector analy i , we have. Car B has both tangential and normal c mponents of acceleration. The magnitude of the normal component is. Thu , from Fig. Refer to the comment made at the end of Example Fl Fl Detennine the velocity of car A if point P on the cable has a peed of 4 mI s when the motor M winds the cable in.

Determine the velocity of block A if end B of the rope i pulled down with aped of 1. Determine the velocity of car B relative to car A. Two plane. A and B are trave lin g with the can tant ve locili hown.

D e lenlline th e magnitud and direction of the ve locity of plane B relative to plane A. At th e in lanl hown car A and B are lraveling at the peeds shown. A and B when thi. Determi ne the di. The crate i being lifted up the inclin d plan using the motor M and the rope and pulley arran gem nl hown.

The pulley arrangemen t shown is de igned for hoi t ing materials. If B rem. D eterm ine the speed of block A if the end of the rop i pulled down wi th a peed of 4 ml. The cylinder C i being lifted u ing th e cable and pulley y. If point A on the cable i. Th e m an pull the boy up to th tree limb by wa lkin g backwa rd a t a consta nt p ed of 1. Th e gir l at p ulls in th e rop horizontally a t a consta nt pe d of 6 ft j.

If he. Neglect the size of the lim b. Whe n the roller i at B, the crate rests on the gro und. Neglect the size of the pulley in t he ca lcul atio n.

Re la te th coordin a te Xc and XA u ing th e proble m geometry, th n lake th e firs t a nd econd lime derivati ve '. D e te rm in e the velocity and acce lera tion of block B a t thi in ta nt. Vertical mo tion of the load i produced by mo ve ment of the pi 'lon at A o n the boom.

D e termin e the. T he cable i. If b lock B i mov ing down with a e locity VfI and ha an acce le rati o n 08, de te rmin e the velocity and accel rati o n of block A in term of th para m l r ShOWll. B are trave lin g ctiv Iy, If B is A maintain a acce lerati o n of curve having a. D etemlin e the velocity of car B re lati ve to car A. At the in tant ' hown, th e plane at A ha. The bicycJi t at B is travel ing at 8.

D e term in e the re lati velocilY and relative accelera tion of A with respect to B at this insta nt. An in tr ument in th e car indicates that the wind i coming from th e cast. D temlin th 'peed and dir ction of th wind. Two boa t leave th e shore at th e same tim e and the directions hown. H ow long afler laving th hore will th boat be ft apart?

D e termin e the r la tive velocilY and relative accel ra tion of car A with re pect to car B at thi in tant. D termin e the ' pe d of A with resp ct to B if A trave ls along the circuJar track , while B travel along th e diam e ter of th e circle.

Compute the termin al con tant ve locity Vr of the rain if it i a sumed to fa ll vertical ly. At th inst. At the in tant. He wishe to cro s th e ft-wide river to point B 30 ft down tream.

While in the water he must not direct him se lf toward point B to reach this point. The hip travel at a con.

Determine the magnitude and direction of the horizontal component of velocity of the make corning from the moke tack a it appear to a pa enger on Ih hip. Determine the con tant peed at which the p layer at B mu. AI o calculate the relative velocity and relative acceleration of the football with respect to B at the instant the catch is made. Player B i 15 m away from A when A tart. Determine the velocity and acceleration of car A relative to car C. Determine the ve locity and acceleration of car 8 relative to car.

Both boats A and B leave the hare at 0 at the same time. If you measured tile time it! The pilo1 tells you the wingspan of her plane and her constan t airspeed. How would you determine the accelera tion of the plane at the moment shown? Usc numerical I'alues and ta ke any necessary measurements from the photo. A position coordinate s specifics the location of the particle on the line. The average velocity is a vector quantity. If the acceleraTion is known to be consta nt. Curvilinear motion "Iollg the path can be resolved into rectilillear motion alollg the x.

TIle equation of the path is used to relate the mot ioll along each axis. It has a constant velocity in the horizolltal d irection. Any two of the three equations for constant acceleration apply in th e vertical directi o n. Curvilinell r Motion II, r I f normal and t: The normal component a.

This component is always in the positive II directio ll. If the path of motion is expressed in polar coordinales. Once the data are substituted into the equations. Absolute De pendent Motion of Two Particles The dependent motion of blocks Ihal are suspe nded from pulleys and cables can be related by the geometry of the syslem.

This is done by first establishi ng position coordinates. Each coordin ate must be directed along the line of motion of a block. The first time derivative of this eq uation gives a relatio nship between the velocities of the blocl.: For planar motion. For sol ution. Kinetics of a Particle: To analyze the accelerated motion of a particle using the equation of motion with different coordinate systems.

Th is law can be verified experimenlally by applying a known unbalanced force F to a particle. Since the force and acceleration are di rectly proportio nal. This positive scalar m is called the mass of the pa rt icle. Being constant du ring any acceleration. If the maS5 of the particle is m, Newto n's second law of motion may be written in ma thema tical form as. The above equillion, which is referred to as the equarion of motion, is une of the most important fo rmula tions in meehanics,- As previously stated , its validity is based solely on experimcTlTal cllidencc, In , however, A Ibert Einstein developed the theo ry of rela tivity and placed limitations on the use of Newton's second law for describing general particle motion, Through experiments it was proven that lime is not an absolute quantity as assumed by Newton: For the most part.

Newton's Law of Gravitational Attraction. Shortly after for mulating his three laws of motion, Newton postulated a law governing the mu t ual att rac tion between any two particles. In mathematical fu rm th is law can be expressed as. In the case of a particle located at o r nea r the surface of the earth.

This fo rce is termed the "weight " and. From Eq. For most engineering calculations g is meas ured at a point on the surface of the earth at sea level. In the SI systcm the mass of thc body is spccificd in kilog rams. Fig, la. As a result. In the FPS system the weight of the body is specified in pounds. The mass is measured in Slugs. It must be calculated, Fig. Therefore,a body weighing Wh en more than one force acts on a particle.

For this more general case. To illustrate 'Ipplication of this equation. We can graphically ,! Ccount for the magnitude and di rection of each force acting on the particle by drawing the particle's f ree-body diagram, Fig. Such are the conditions of sialic equilibrium. Newton's first law of motion. Inertial Reference Frame. When applying!

In this way. Wlm e from allY reference of this type. Such a frame of reference is commonly known as a Newtonian or inerlial reference frame, Fig. Wh en studying the motions of rockets ,md satellites.

Even though the earth both rotates about its own axis and revolves about the sun, the accele rations created by these rotations are re latively small and so they can be neglected for most applications. Thi s method of apptication , which "ill nOl be used in th is lexl. We are all familiar with the sensation one feels when silling in a car that is subjected to a forward acceleration. Often people think this is caused by a "force" which acts on thcm and tends to push them back in their seats; however.

Consider the passenger who is strapped to the seat of a rocket sled. Pro vided the sled is at rest o r is moving with constant velocity. When the thrust of the rocket engine causes the sled to accele rate. In th e photo. Upon dcccleration thc force of the scatbelt F' tcnds to pu! No force is pu! TIle e quation of motion will now be e xtended to include a system of particles isolated within an enclosed region in space.

In particular. A t the instant considered. The inrernal force. The free-body and kinetic di'lgrams for Ihe ilh particle ,Ire shown in Fig. I b. Applying the equation of motion,. When Ihe equation of motion is applied to each of the other particles of the system.

TIle summation of the internal forces, if carried out. D ifferentiating this equation twice with respect to time. Since in rea! An inertial frame of reference does not rotate, rather its axes either translate with constant velocity or are al rest.

Mass is a propert y of matter that p rovides a quantitative measure of its resistance to a change in velocity. It is an absolute quantity and so it does not change from one location to another. Weigh t is a force that is eilused by thc carth's gmvitation. When a particle moves relative to an inertial x.

Z fram e of refe rence, the fu rces acting on the panicle. Applying the equation o f motion. For this equation In be satisfied. Conseque ntl y, we may write the follo wing three scalar equations: Free-Body Diagram. Select the inertial coordinate system. Most often. Once the coordinates are established. Drawing this diagram is very imporUmt since it provides a gra phical represent ation tha t accounts for llilhe forces 'i F which act on the particle.

The direction and sense of the particle's acceleration a should also be established. If the sense is unknown. TIle acce le ration may be re prese nted as the kinetic diagram. Identify the unknowns in the proble m. If the forces can be resolved directly from the free-body diagram. If the geometry of the prublem appears complicated. Remember that Ff always acts on the free-body diagram such that it o pposes the motion of the particle relative to the surface it contacts.

If the particle is VII the verge of relative motion. If thc particle is connected 10 an elastic spring having ncgli gible mass. He re k is the sprin g's stiffness measured as a force per unit length. If the velocity or position of the particle is 10 be found.

If acceleraTion is a function of displacement. If the problem involvcs the dependcnt motion of several particles. If the solution for an unknown vec tor component yields a negative scalar, it indicates that the component acts in the dircction opposite to that which was assumed. The kg crate shown in Fig. If you continue browsing the site, you agree to the use of cookies on this website.

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