Chapter 7 LINEAR MOMENTUM

Transcription

1 Chapter 7 LINEAR MOMENTUM Conceptual Questions 1 The likelihood of injur resulting fro juping fro a second floor window is priaril deterined b the aerage force acting to decelerate the bod (a) The deceleration tie interal for a person landing stiff legged on paeent is er short The ipulseoentu theore tells us that the aerage force acting on the person s feet ust therefore be er large such a person is likel to incur injuries (b) Juping into a priet hedge increases the tie interal oer which the bod decelerates This decreases the aerage force on the person s libs and therefore decreases the likelihood of injur (c) Juping into a firefighter s net is the best option of the three The net stretches downward, graduall bringing the person to rest Additionall, the firefighters lower the net with their hands as the person lands to further lengthen the tie interal during which the person is brought to rest (a) A bod s oentu change is equal to the ipulse that has acted on it Ipulse is defined as the product of the aerage force acting on a bod and the tie interal oer which it acts the bodies therefore eperience the sae ipulse and so hae equal oentu changes (b) The change in a bod s elocit is defined as the ratio of the change in its oentu to its ass the less assie bod therefore incurs a larger elocit change (c) The acceleration of a bod is defined as the ratio of the force acting on it to its ass the less assie bod therefore has the larger acceleration 3 The uzzle speed is deterined b the change in the bullet s oentu The ipulse-oentu theore tells us that this oentu change is deterined b the ipulse acting on the bullet The force acting on the bullet due to the epanding hot gases is roughl constant throughout the uzzle A shorter uzzle produces a shorter tie interal oer which the bullet is accelerated b the firing force This results in a saller ipulse and therefore a saller oentu change thus producing lower bullet elocities 4 After the eplosion, each piece of the firecracker has a oentu ector associated with it that points in the direction of its otion The law of conseration of linear oentu tells us that the ector su of the oentu of all the pieces of the firecracker ust equal the initial oentu of the whole firecracker in this case, both the initial and the final net oentu ectors equal zero 5 The law of the conseration of linear oentu states that in the absence of eternal interactions, the linear oentu of a closed sste is constant Floating in free space, the astronaut and the wrench for a closed sste free fro interactions with other bodies If the astronaut throws the wrench in the direction opposite the ship, conseration of oentu dictates that he ust in turn oe toward the ship 6 The horizontal coponent of the golf ball s oentu is consered since no eternal force acts on the ball in the horizontal direction The ertical coponent of the ball s oentu is not consered howeer because the Moon s graitational force interacts with it and changes its oentu 371

2 Chapter 7: Linear Moentu Phsics 7 In an elastic collision between the haer and nail, the kinetic energ of the sste is consered while in a perfectl inelastic collision, the greatest percentage of the kinetic energ is lost The energ lost b the sste in a perfectl inelastic collision is used to do the work required to bring the haer and nail together In an elastic collision, this work is aailable to drie the nail into the wood the total work aailable to drie the nail is therefore greater for an elastic collision Thus, for equal applied forces, the haer will drie the nail further into the wood if the collision is elastic 8 The oentu of the squid (including the water inside its cait) coprises a sste for which oentu is consered The eans the oentu of the squid (plus water) ust be the sae before and after soe of the water has been ejected When the squid epels soe water, the water gains oentu in the direction it is being epelled To consere oentu, the squid ust gain an equal aount of oentu in the opposite direction, propelling it forward Siilarl, a rocket engine epels ehaust fro burning fuel to propel itself forward 9 First law: The oentu of an object is constant unless acted upon b an eternal force Second law: The net force acting on an object is equal to the rate of change of the object s oentu Third law: When two objects interact, the changes in oentu that each iparts to the other are equal in agnitude and opposite in direction 10 Noting that the (translational) kinetic energ can be written as p /( ), and that both objects hae the sae kinetic energ, it is eident that the object with the greater ass has the larger agnitude of oentu 11 The woan s center of ass is not necessaril 080 aboe the floor, because her ass is not necessaril distributed uniforl with height Norall, the upper bod of a person is ore assie than the lower bod and thus we would epect the woan s center of ass to be slightl higher than The frictional force of the road on the tires supplies the eternal force to change the biccle s oentu Changes in the biccle s kinetic energ do not require an eternal force For eaple, the rider could throw her helet awa hard, increasing both her and the helet s speed The kinetic energ of the sste (biccle, rider, and helet) would increase, while the oentu would reain the sae Note that the work-energ theore (total work done equals change in kinetic energ) cannot be used here, because the internal structure of the sste cannot be ignored 13 An ipulse ust be supplied to the egg to change its oentu and bring it to rest A good strateg is to ake the tie interal oer which the stopping force is applied as large as possible This will reduce the agnitude of the force required to stop the egg One should therefore attept to catch the egg with a swinging otion, oing the hand backwards as it is being caught, to bring it to rest as slowl and gentl as possible 14 The collisions of the balls in the eecutie to are nearl perfectl elastic The kinetic energ of the sste just before and after a collision ust therefore be the sae This is the reason we neer see three balls oing awa after a collision in which two balls were initiall pulled back and released such an eent would not consere kinetic energ 15 According to the ipulse-oentu theore, the change in oentu of the baseball is equal to the ipulse it receies fro the bat Ipulse is equal to the aerage force ties the tie interal oer which the force is applied To gie the ball the greatest possible oentu, one should attept to aiize the aount of tie during which the force is being applied 16 Jere has it right B oentu conseration, Micah needs to throw the balls forward if he wants to propel hiself backward, but the balls need not strike an surface You can also consider Newton s third law and see that it is the force b the balls on Micah s hand that pushes Micah backward 17 Darl has done his hoework If he falls when rock clibing, his rope will stretch and stop hi ore graduall than the rope Mar wants to bu In a fall, the cliber s oentu ust go fro soe initial alue to zero If the tie oer which the oentu is decreased to zero is longer, the aerage force deliered b the rope is saller 37

3 Phsics Chapter 7: Linear Moentu Probles 1 Strateg Use the definition of linear oentu Solution Find the agnitude of the total oentu of the sste ptotal p1 p 1 ( 1 ) [ 1 ( 1)] 0, so the agnitude is 0 Strateg Use the definition of linear oentu Solution Find the oentu of the autoobile W 9800 N (35 s south) p kg s south g 980 s 3 Strateg and Solution Ipulse F t, so the SI unit is N s kg s s kg s p, so the SI unit is kg s Therefore, the SI unit of ipulse is the sae as the SI unit of oentu 4 Strateg Use the ipulse-oentu theore Solution Find the final speed of the cue ball Fat (4 N)(008 s) p pf pi f (0) Fat, so f 4 s 016 kg 5 Strateg Add the oenta of the three particles Solution Find the total oentu of the sste ptot p1 p p north south 33 north ( 11 33) north (30 kg)(30 s) (40 kg)(50 s) (70 kg)(0 s) north 3 kg s north 6 (a) Strateg For a ratio of the agnitudes of the final and initial oenta Solution Copute the ratio pf f f 600 i h p 00 i h i i i 300 (b) Strateg For a ratio of the final and initial kinetic energies Solution Copute the ratio 1 K f f f K 1 i i i 7 Strateg The initial oentu is toward the wall and the final oentu is awa fro the wall Solution Find the change in oentu p pf pi f i ( f i) (50 kg)( 0 s 0 s) 0 kg s, so p 0 kg s in the -direction 373

4 Chapter 7: Linear Moentu Phsics 8 Strateg The final and initial elocities are the sae, since air resistance is ignored Use the definition of the 1 linear oentu Use Eq it g ( t ) to find the initial speed Let up be the positie direction Solution Find the initial speed it g( t), so i g t Find p 1 p p f pi ( f i) ( i i) gt gt (30 kg)(980 s )(34 s) 1010 kg s So, p 1010 kg s downward 9 Strateg Use the definition of linear oentu Let up be the positie direction Solution f f i gt g t, since the object starts fro rest Find p p p f pi ( f i) ( gt0) gt (30 kg)(980 s )(34 s) 1010 kg s, so 1010 kg s downward p 10 Strateg Use the ipulse-oentu theore Solution Find the aerage force p (500 kg)(30 s 0) Fa 75 N t t 00 s The force necessar is 75 N in the direction of the sled s elocit 11 Strateg Use the ipulse-oentu theore Let the forward direction be positie Solution Find the tie interal for which the engine ust be fired a a p (3800 kg)(1110 s 610 s) t F F 1810 N 30 s 374

5 Phsics Chapter 7: Linear Moentu 1 (a) Strateg Use the coponent ethod of subtracting ectors Solution Copute the agnitude of the change in oentu p and p p ( ) ( ) (015 kg) [0 ( 0 s)] (15 s 0) 38 kg s Find the angle of the change in oentu 1 1 tan p tan tan p 0 The change in oentu of the baseball is 38 kg s at 37 aboe the horizontal direction opposite i p f p i (b) Strateg Use the ipulse-oentu theore Solution Find the aerage force of the bat on the ball p 375 kg s Fa 75 N, so F 75 N in the sae direction as p t 0050 s 13 Strateg Use the ipulse-oentu theore Let the positie direction be in the direction of otion Solution Find the aerage horizontal force eerted on the autoobile during breaking p ( 3 f i) (1010 kg)(0 300 s) 3 Fa 6010 N t t 50 s So, F 3 a 6010 N opposite the car s direction of otion 14 Strateg Use the ipulse-oentu theore Solution (a) Copute the changes in oenta for each direction p 0 and p F t north east a east east Find the agnitude and direction of f Fat (15 N)(40 s) f north east north (15 s) 5 s and 30 kg 1north 115 s tan tan 37 north of east, so f 5 s at 37 north of east 0 s east 15 /s N 15 N (b) Let + be north and + be east Copute the change in oentu p Fa t (15 N)(40 s) 60 kg s The entire change in oentu is due to the force, so p 60 kg s east p i Δp p f 375

6 Chapter 7: Linear Moentu Phsics 15 (a) Strateg Use the definition of linear oentu Use a to find the speed after the fall 376 f i Solution Find the initial speed, which is the final speed after the fall f i f 0 g gh, so f gh If up is positie, gh down gh up i p ( f i ) 0 gh gh (600 kg) (980 s )(80 ) 750 kg s, so p 750 kg s upward (b) Strateg The ipulse on the net is equal to the bo s weight ties bo due to the net, p t plus the change in oentu of the Solution Find the ipulse on the net gt downward p (600 kg)(980 N kg)(040 s) downward 750 kg /s downward 990 N s downward (c) Strateg Use Eq (7-3) Solution Find the aerage on the net due to the bo p 990 kg /s downward Fa 500 N downward t 040 s 16 Strateg The ipulse is equal to the area under the graph Use the ipulse-oentu theore Let the positie direction be to the right Solution Each rectangle of the grid is equal to (100 N)(00010 s) 010 kg s The area can be diided easil into three right triangles and one rectangle Thus, there are 1 (7)(4) 1 (6)() 1 (8)(6) (6)(4) 68 rectangles under the graph and the agnitude of the ipulse is p 68(010 kg s) 68 kg s The ipulse is opposite the direction of otion of the initial elocit Copute the final speed p p 68 kg s f i, so f i 30 s 9 s 0115 kg 17 (a) Strateg Use conseration of energ to find the speed with which the pole-aulter lands on the padding Solution 1 K U gh, so gh (980 s )(60 ) 1084 /s 11 s (b) Strateg The padding eerts an upward force on the pole-aulter while grait continues to eert a downward force Use the ipulse-oentu theore and let F a represent the aerage force eerted b the padding Solution p Fnett, and Fnet Fa g, so p ( f i) [0 ( gh)] Fa Fnet g g g g t t t gh 1084 /s Fa g 600 kg 980 s 1900 N t 050 s

7 Phsics Chapter 7: Linear Moentu 18 Strateg Use conseration of oentu Solution Find the recoil speed of the rifle b kg prf pbf r rf b bf pri pbi 00, so rf bf (80 s) 18 s 45 kg 19 (a) Strateg Right after the collision, the bullet and baseball cobination ust hae the sae oentu as the bullet had just before it stuck the baseball Solution Before the collision, the oentu of the bullet is p i bullet bullet After the collision, the oentu of the bullet and baseball cobination is pf ( bullet baseball) f pi Thus, the speed of the bullet and baseball cobination right after the collision was pi bulleti (0030 kg)(00 s) f 33 s 0030 kg 015 kg bullet baseball bullet baseball r (b) Strateg Use conseration of energ to deterine the work done b air resistance on the bullet and baseball cobination Solution Deterine the work done b air resistance Let up be the positie direction Wtotal Wc Wnc U Wair K, so 1 1 Wair K U 0 g (018 kg) (33333 s) (980 s )(37 ) 3473 J Wair 3473 J Since Wair Fair, a, Fair, a 094 N Therefore, the aerage force of air resistance 37 was 094 N down 0 Strateg Use conseration of oentu Solution Find the recoil speed of the subarine t 50 kg psf ptf s sf t tf psi pti 0 0, so sf tf (1000 s) 0010 s kg 1 Strateg Use conseration of oentu Solution Find the recoil speed of the thoriu nucleus p 0 p, so if n = nucleus and p = particle, i f p 40 u 8 5 pn pp nn pp 0, so n p 0050(99810 s) 610 s n 34 u Strateg Use the law of conseration of linear oentu to deterine the speed Dash ust throw the balls Solution According to the law of conseration of linear oentu, Dash and his skateboard will oe backward with linear oentu equal in agnitude to the agnitude of the cobined oentu of the balls Find the speed of the balls DD (60 kg)(050 s) pb 3 bb DD pd, so b 100 s (4 ph) 3b 3(010 kg) Since 4 ph is faster than an huan can throw a ball, Dash will not succeed s 377

8 Chapter 7: Linear Moentu Phsics 3 Strateg Use the law of conseration of linear oentu to deterine the astronaut s speed Solution According to the law of conseration of linear oentu, the astronaut will oe toward the ship with linear oentu equal in agnitude to the agnitude of the cobined oentu of the objects thrown Find the speed of the astronaut after he throws the allet p objects p w p s p w w s s p A A A, so ww ss A A (07 kg)(50 s) (080 kg)(80 s) (1 kg)(60 s) 030 s 58 kg 4 Strateg Use conseration of energ to deterine the skier s speed and oentu just before he grabs the backpack Then, use conseration of linear oentu to find his new speed after he grabs the backpack Finall, fro Chapter 4, use the equations for otion with a changing elocit Solution Use conseration of energ to find the speed of the skier just before he grabs the backpack 1 1 E K U K f Ki Uf Ui ss 00sgh ss sgh 0, so s gh Use conseration of linear oentu to find his new speed after he grabs the backpack s s s gh pi ss pf ( s b), so s b s b Now, find the tie it takes for an object to fall 0 fro rest i t g( t) (0) t g( t) g( t), so t g The skier will trael a horizontal distance of s gh s h (65 kg) ( 0 )(50 ) t g 65 kg 0 kg s b s b 5 Strateg Use conseration of oentu 48 Solution Find the recoil speed of the railroad car pi 0 pf, and since we are onl concerned with the horizontal direction, we hae: s 98 kg c c s s, so c s (105 s)cos s kg 6 Strateg Use conseration of oentu c Solution Find the ass of the an and the car p 0 p, and since we are onl concerned with the horizontal direction, we hae: i f b (173 s)cos300 cc bb, so c b (0010 kg) 1500 kg 3 c 1010 s 378

9 Phsics Chapter 7: Linear Moentu 39 Strateg Use conseration of oentu Let the positie direction be to the right Solution Find the final elocit of the heliu ato He Hef O Of He Hei O Oi, so O( Oi Of) HeHei Hef He O 30 u ( Oi Of ) Hei 41 s 456 s618 s 70 s 400 u He Thus, the elocit of the heliu ato after the collision is 70 s to the right 40 Strateg Linear oentu is consered, so p f p i Solution Find the change in speed of the car p ( ) p, so, and car f car cla f i car i f i car cla car car 10 g f i i i 1 i 1 (075 s) 015 s car cla car cla 10 g 300 g 41 Strateg Use conseration of oentu Solution (a) The collision is perfectl inelastic, so 1f f f Find the speed of the two cars after the collision 1i 10 s 11ii 1i 40 (0) 11f f f40 f, so f 00 s (b) The cars are at rest after the collision, so 1f f 0 1i 10 s 1i 40i 0, so i 05 s The initial speed was 05 s Strateg Use conseration of oentu The block is initiall at rest, so i 0 Let east be in the +direction Solution Find the final elocit of the block 11f f 11ii 11i(0), so 1( 1i 1f) 000 kg f 000 s ( 1000 s) 30 s 0 kg Thus, block 30 s east 43 Strateg Use conseration of oentu The collision is perfectl inelastic, so 1f f f Also, the block is initiall at rest, so i 0 379

10 Chapter 7: Linear Moentu Phsics Solution Find the speed of the block of wood and the bullet just after the collision 1 1f f ( 1 ) f 1 1i i 1 1i (0), so kg f 1i (1000 s) 50 s 0050 kg 095 kg 1 44 Strateg The collision is perfectl inelastic since the bullet ebeds in the wood Friction does negatie work on the block and bullet cobination Use Newton s second law and conseration of oentu Solution Let the +-direction be in the direction of otion Find the acceleration of the block and bullet due to friction F N ( bullet block ) g 0, so N ( bullet block ) g F fk k N k ( bullet block ) g ( bullet block ) a, so a k g Find the initial speed of the block and bullet (just after the collision) f i i k i k 0 a g, so g Use conseration of oentu to find the speed of the bullet just before its collision with the block bulletbullet ( bullet block ) ( bullet block ) k g, so ( bullet block ) k g (0 kg) (0400)(980 s )(150 ) bullet 350 s 000 kg bullet 45 Strateg Use conseration of oentu Solution Find the total oentu of the two blocks after the collision p p1 pf pi p1i p1f p1f pf p1i pi ( 1) f 11i i p (0 kg) 10 s (10 kg)(0) 0 kg s p f 1i f k N ( bullet + block )g Since p 1i was directed to the right, and pf p1i, the total oentu of the two blocks after the collision is 0 kg s to the right 46 Strateg The collision is perfectl inelastic, so 1f f f Use conseration of oentu Let the positie direction be the initial direction of otion Solution Find the speed of the an (1) just after he catches the ball () 1 1f f 1 f f 1 1i i 1 (0) i, so 00 kg f i (5 s) 0066 s 75 kg 00 kg 1 47 Strateg Use conseration of oentu Let the positie direction be in the initial direction of otion Solution Find the speed of the Volkswagen after the collision p p, so V V Vf V Vi B B B 3 3 VViBB (1010 kg) 5 s (010 kg)(33 s 4 s) Vf 3 V 1010 kg 43 s 380

11 Phsics Chapter 7: Linear Moentu 48 Strateg Use conseration of linear oentu Since the collision is perfectl elastic, kinetic energ is consered Solution The 100-g ball is (1) and the 300-g ball is () Note that ii 11i(0) 11i 11ff, so 1i 1f f 1f 3 f i 1 (0) 1 1 1i 1 1 1f f 1 1f (3 1 ) f, so 1i 1f 3 f Substitute for 1i 1f 3f 1f 61f f 9f 1f 3 f, so 61f f 6 f, or 1f f Find the final elocities of each ball 1 1 1i 1f 3f f 3f f, so f 1i (500 s) 50 s Since 1f f, 1f 50 s So, the 300-g ball oes at 50 s in the + -direction ball oes at 50 s in the -direction and the 100-g 49 Strateg Use conseration of oentu Let the positie direction be the initial direction of otion Solution Find the speed of the 50-kg bod after the collision 11ff 11ii, so 1( 1i 1f) i (10 kg) 100 s ( 50 s) (50 kg)(0) f 50 kg 30 s 50 Strateg The collision is perfectl inelastic, so 1f f f Use conseration of oentu Let the positie direction be the initial direction of otion Solution Find the speed of the cobination 1(0) i 30 kg 1 ff 1 1ii, so f (80 s) 48 s 0 kg 30 kg 1 51 Strateg The spring iparts the sae (in agnitude) ipulse to each block (The sae agnitude force is eerted on each block b the ends of the spring for the sae aount of tie) So, each block has the sae final agnitude of oentu (The initial oentu is zero) Solution Find the ass of block B BB AA, so A da t da 10 B A A A (060 kg) 00 kg d t d 30 B B B p A p B 381

12 Chapter 7: Linear Moentu Phsics 5 (a) Strateg Use conseration of oentu Let the +-direction be to the right Solution Find the final elocit of the other glider, so 050 s 050 s 130 s 030 s 1f f i i f i i 1f So, the elocit of the other glider is 030 s to the left (b) Strateg For a ratio of the final to the initial kinetic energies Solution Copute the ratio 1 1 K 1f f f 1f f (130 s) (030 s) 36 K 1 i i i (050 s) The final kinetic energ is greater than the initial kinetic energ The etra kinetic energ coes fro the elastic potential energ stored in the spring 53 Strateg The collision is perfectl inelastic, so 1f f The block is initiall at rest, so i 0 and 1i i Use conseration of oentu Solution Find the speed of the bullet and block sste bul ( bul blk ) buli blk (0), so i bul blk Deterine the tie it takes the sste to hit the floor 1 1 h h it g( t) 0 g( t), so t g Find the horizontal distance traeled h = 1 h 0010 kg (1 ) t t 4000 s 049 bul i i bul blk g 0010 kg 40 kg 980 s 54 Strateg Use conseration of oentu Ki Kf, since the collision is elastic Solution Show that the final speed of each object is the sae as the initial speed 11ff 11i i and p1i pi So, 11i i and 11f f 0, or 11f f 1 1 1i 1 i 1 1 1f 1 f, so 1 1i i 1 1f f Eliinate 1i and 1f 1 i i 1 f f 1 1 i f 1 1 i f Δ 38

13 Phsics Chapter 7: Linear Moentu Therefore, the initial and final speeds of object are the sae Eliinate i and f i 1i 1 1f 1f i 1 1f 1i 1f Therefore, the initial and final speeds of object 1 are the sae 55 Strateg Use conseration of linear oentu Since the collision is perfectl elastic, kinetic energ is consered as well Solution Let the +-direction be in the original direction of otion of the 0-kg object The 0-kg object is (1) and the 60-kg object is () Note that ii 11i(0) 11i 11f f, so 1i 1f f 1f 3 f i 1 (0) 1 1 1i 1 1 1f f 1 1f (3 1 ) f, so 1i 1f 3 f Substitute for 1i 1f 3f 1f 61f f 9f 1f 3 f, so 61f f 6 f, or 1f f Find the final speed of the 60-kg object 1 1 1i 1f 3f f 3f f, so f 1i (10 s) 50 s 56 Strateg Look at the collision in its center of ass frae Assue a one-diensional collision The initial elocities are 1i and i The asses are 1 and Solution Transfor the initial elocities to the frae b subtracting fro each 1i 1i i i According to the result fro Proble 54, the final speeds of the objects ust be the sae as the initial speeds, but the final and initial elocities are oppositel directed since the objects rebound after colliding Therefore, 1f 1i 1i f i i Transfor back to the original frae of reference 1f 1f 1i f f i The relatie speed after the collision is 1f f 1i i 1i i, which is the relatie speed before the collision 57 Strateg Use conseration of oentu Let each of the first two pieces be 45 fro the positie -ais (one CW, one CCW) Solution Find the speed of the third piece Find 3 383

14 Chapter 7: Linear Moentu Phsics , so 3 1 cos 45 cos( 45 ) p p p Siilarl, 3 1 sin 45 sin( 45 ) 0, so s 170 s 58 Strateg Use conseration of oentu Solution Find Bf pi MAi pf MAf MBf, so Bf Ai Af Find Bf pi 0 pf MAf MBf, so Bf Af Calculate Bf Bf ( Ai Af) ( Af) 60 s 10 s 0 s 54 s 59 Strateg Use conseration of oentu Refer to Practice Proble 711 Solution (a) Find the oentu change of the ball of ass 1 1 p1 p i f (0 f) f 51 i cos( 369 ) i 1 p1 p i f 5 1 (0 f) 5 1 f 51 i sin( 369 ) i (b) Find the oentu change of the ball of ass p p1 1( 1i 1f) 1( i 0) 1i p p1 1( 1i 1f) 1(0 1) 11 1(0751 i) 07511i The oentu changes for each ass are equal and opposite 60 Strateg Use conseration of oentu Let right be + and + be in the initial direction of the puck Solution Find f 1f f 1i i 00, so f 1f Find f 0, so 1f f 1i i 1i f 1i 1f 1i 1f θ1 =

15 Phsics Chapter 7: Linear Moentu Calculate f f f f ( 1f) ( 1i 1f) ( 1f sin 1 ) ( 1i 1f cos 1 ) 036 ssin s 036 scos37 07 s Calculate the direction of the second puck 1 f 1 1f ssin 37 tan tan tan 53 to the left f 1i 1f 045 s 036 scos37 Thus, f 07 s at 53 to the left 61 Strateg Use conseration of oentu Solution Find f in ters of 1f 1f f 1f sin 1 f sin 1i i 0 0, so sin1 sin 600 f 1f 1f 173 1f sin sin( 300 ) 1i 1f f 6 Strateg The collision is perfectl inelastic, so the final elocities of the blocks are identical Use conseration of oentu Solution Find the initial speed of block B pi AAi BBi 0 BBi BBi pf ( A B) f, ( A B) f ( A B) f cos so Bi Bi Copute Bi B B (0 g 300 g)(313 s)cos(18045 ) Bi 40 s 300 g f 45 Ai Bi N Thus the initial speed of block B was 40 s 63 Strateg Use conseration of oentu Let + be along the initial direction of the projectile Solution Find the agnitude of the oentu of the target bod after the collision Find p f p pf pi pf 0 p1 p1i p1f i f ( i f cos ), so pf ( i f cos ) Find p f p pf pi pf 0 p1 p1i p1f 0 f sin, so pf f sin Calculate p f p f p f p f ( i f cos ) f sin (0 kg) 50 s30 scos s sin kgs 385

16 Chapter 7: Linear Moentu Phsics 64 (a) Strateg The collision is perfectl inelastic, so the final elocities of the cars are identical Use conseration of oentu Solution Let the 1500-kg car be (1) and the 1800-kg car be () p 1 i 11i i 11i 0 pf ( 1) f, so f 1i 1 p i 11i i 0 i pf ( 1) f, so f i 1 Copute the final speed 11i i [(1500 kg)(17 s)] [(1800 kg)( 15 s)] f f f kg 1800 kg 11 s Copute the direction 1 f 1(1800 kg)( 15 s) tan tan 47 f (1500 kg)(17 s) Thus, the final elocit of the cars is 11 s at 47 S of E 1i i N (b) Strateg Find the change in kinetic energ Solution K Kf Ki ( 1) f 11i i (1500 kg 1800 kg)(1154 s) (1500 kg)(17 s) (1800 kg)(15 s) 10 kj Thus, 10 kj of the initial kinetic energ was conerted to another for of energ during the collision 65 Strateg The collision is perfectl inelastic, so the final elocities of the cars are identical Use conseration of oentu 386

17 Phsics Chapter 7: Linear Moentu Solution Let the 1700-kg car be (1) and the 1300-kg car be () p 1 i 11i i 11i 0 pf ( 1) f, so f 1i 1 11i i pi 11i i pf ( 1) f, so f 1 Copute the final speed and the direction f 11i 11i i f f [(1700 kg)(14 s)cos 45 ] [(1700 kg)(14 s)sin 45 (1300 kg)( 18 s)] 1700 kg 1300 kg 1 f 1(1700 kg)(14 s)sin 45 (1300 kg)( 18 s) tan tan 1 (1700 kg)(14 s)cos 45 f 1 1 N 1i 60 s i Thus, the final elocit of the cars is 60 s at 1 S of E 66 Strateg Use conseration of oentu Solution Find the coponents of the deuteron s elocit after the collision Find df n n ni d di n i 0 n nf d df 0 d df, so df i d Find df i 1 n n ni d di 00 n nf d df n d df, so df i 3 3 d Find the coponents, df and df i i / 3 n 1 n n 1 n i i ( df, df) i, i i, i, d 3 d n 3 n 3 67 Strateg Use conseration of linear oentu Solution Find Bf pi Ai pf Af Bf, so Bf Ai Af Find Bf pi Ai 0 pf Af Bf, so Bf Af Copute the final speed of puck B Bf ( Bf) ( Bf) ( Ai Af) ( Af) Ai 60 Af [0 s (10 s)cos60 ] [ (10 s)sin 60 ] 17 s Copute the direction of puck B 1 Bf 1 (10 s)sin 60 tan tan 30 0 s (10 s)cos60 Bf 387

18 Chapter 7: Linear Moentu Phsics Thus, the speed and direction of puck B after the collision is 17 s at 30 below the -ais 68 Strateg The collision is perfectl inelastic, so the final elocities of the acrobats are identical Use conseration of oentu Solution Let the first acrobat be (1) and the second acrobat be () p p ( ), so ( ) ( ) 0 /s 30 /s i 1 1i i f 1 f f 1 1i i 1 pi 11i i pf ( 1) f, so f ( 11i i) ( 1 ) 0 10 Copute the final speed ( 11i i) ( 11i i) f f f 1 [(60)(30 s)cos10 (80)(0 s)cos160 ] [(60)(30 s)sin10 (80)(0 s)sin160 ] s Copute the direction 1 f 1 (60)(30 s) sin10 (80)(0 s) sin160 tan tan 73 f (60)(30 s) cos10 (80)(0 s) cos160 Thus, the final elocit of the acrobats is 064 s at 73 aboe the + -ais 69 Strateg Use conseration of oentu 388

19 Phsics Chapter 7: Linear Moentu Solution Let swallow 1 and its coconut be (1) and swallow and its coconut be () (before the collision) After the collision, let swallow 1 s coconut be (3), swallow s coconut be (4), and the tangled-up swallows be (5) 30 N pi pf , so pi pf , so Copute the final speed of the tangled swallows, [ ( 33 44)] ( ) 5 [(080 kg)(13 s)cos 60 (070 kg)(14 s)cos 60 ] [(107 kg)(0 s) (09 kg)( 15 s) (080 kg)(13 s)sin 60 (070 kg)(14 s)sin 60 ] 070 kg 00 kg 0 s Copute the direction 1 5 tan 5 1 (107 kg)(0 s) (09 kg)( 15 s) (080 kg)(13 s)sin 60 (070 kg)(14 s)sin 60 tan (080 kg)(13 s)cos 60 (070 kg)(14 s)cos 60 7 Since 5 0 and 5 0, the elocit ector is located in the second quadrant, so the angle is fro the positie -ais or 18 west of north Thus, the elocit of the birds iediatel after the collision is 0 s at 18 W of N 70 Strateg Use conseration of oentu The collision is perfectl inelastic, so 1f f f Solution Find the speed of the sled once the book is on it 11f f ( 1) f 11ii 1i0, so 1 50 kg f i (10 s) 083 s 50 kg 10 kg 1 71 Strateg Use conseration of oentu The collision is perfectl inelastic, so 1f f f Solution Find the speed of the cars just after the collision 11f f ( 1) f 11ii 1i0, so g kn f i 170 s 10 s ( ) g 136 kn 90 kn 1 7 Strateg Use Eqs (7-10) and (7-11) Solution Find the elocit of the center of ass of the sste p M so , 389

20 Chapter 7: Linear Moentu Phsics (30 kg)(90 s) (50 kg)( 10 s) (0 kg)(5 s) 37 s 30 kg 50 kg 0 kg 0, so 37 s in the + -direction 73 Strateg Use the definition of linear oentu Solution Find the agnitude of the total oentu of the ship and the crew ptot tot (010 kg 4810 kg)(10 10 s) 5010 kg s 74 Strateg Use the definition of linear oentu and the ipulse-oentu theore Solution (a) Copute the agnitude of the change in oentu of the ball p pf pi f i ( f i) (0145 kg) 37 s 41 s 11 kg s (b) Copute the ipulse deliered to the ball b the bat Ipulse p 11 kg s (c) Copute the agnitude of the aerage force eerted on the ball b the bat p 1131 kg s Fa 38 kn t s 75 Strateg Use the ipulse-oentu theore Solution Find the aerage force eerted b the ground on the ball p Fa t 54 /s 53 /s ( ) ( ) 18 t 0060 kg 53 scos18 54 scos( ) 53 ssin18 54 ssin( ) 34 N 0065 s 76 Strateg The center of each length is its center of ass Use the coponent for of the definition of center of ass Solution Find the location of the center of ass of the rod (0 50 c 100 c) 500 c 3 390

21 Phsics Chapter 7: Linear Moentu Thus, (, ) (500 c, 667 c) (50 c 100 c 50 c) 667 c 77 Strateg The center of ass of each block is its center Add up the indiidual center of ass coponents to find the coponents of the center of ass of the block structure Solution (10 in) 5(0 in) 4(30 in) 0 in (0) 4(10 in) 0 in 30 in 075 in z13z 9z13z 9(0) 3(10 in) z 05 in The center of ass of the block structure is located at (0 in, 075 in, 05 in) 78 Strateg Use the ipulse-oentu theore Solution Copute the force eerted b the strea on a person in the crowd p Fa (4 kg s)(17 s) 410 N t t t 79 Strateg Use the ipulse-oentu theore Solution Copute the aerage forces iparted to the two gloed hands during the catches k h 10 1 h k a k h 10 1 h p k 3600 s Ineperienced: Fa (014 kg) 5000 N t t 3 10 s 3600 s Eperienced: F (014 kg) 500 N 1010 s 80 Strateg The fl splatters on the windshield, so the collision is perfectl inelastic ( fl, final car, final f ) Use conseration of oentu Let the positie direction be along the elocit of the autoobile Solution (a) Copute the change in oentu p p ( ) ( 0) (0110 kg)(100 k h) 001 kg k h car fl fl fl, f fl, i fl car, i So, the change in the car s oentu due to the fl is 001 kg k/h opposite the car s otion (b) Copute the change in oentu pfl pcar 001 kg k h, or 001 kg k h along the car s elocit (c) Copute the nuber of flies N required to slow the car carcar (1000 kg) 1 k h 5 Npfl car car, so N 10 flies p 001 kg k h fl 391 3

22 Chapter 7: Linear Moentu Phsics 81 (a) Strateg The initial oentu of the baseball is p i i The final oentu is zero Solution Copute the change in oentu p pf pi 0 i i (015 kg)(35 s) 53 kg s Thus, the change in oentu was 53 kg s opposite the ball s direction of otion (b) Strateg and Solution According to the ipulse-oentu theore, the ipulse applied to the ball is equal to the change in the oentu of the ball, or 53 kg s opposite the ball s direction of otion (c) Strateg Use the ipulse oentu theore Solution Since the acceleration is assued constant, the tie it takes for the ball to coe to a coplete stop is t a Copute the aerage force applied to the ball b the catcher s gloe p p 35 s 55 kg s opposite the direction of otion Fa a t kn opposite the ball s direction of otion 8 Strateg Use conseration of oentu The collision is perfectl elastic, so K i K f Also, 1i 1i and 1f 1f Solution Find the speed of the target bod after the collision -direction: 1 1f f 0 f 1 1i i 1 1i 0 -direction: 00, so 11f f 11i i f 11f Square the results and add 1f = 60 /s 1i = 80 /s 1 f f f 1 1i 1 1f 1 1i 1 1f, so f ( 1i 1f ) Calculate the kinetic energies 1 11f 1 f 1 11i 1 1 i 11i 0, so 11f f 11i 39

23 Phsics Chapter 7: Linear Moentu Thus, 1 f 1 ( 1i 1f ) ( 1i 1f ) and 1i 1f 1 1i 1f Fro the kinetic energies, 1 1 ( 1i 1f ) ( 1i 1f ) [(80 s) (60 s) ] f ( 1i 1f ) 8 s 1i 1f 1i 1f (80 s) (60 s) 1 1i 1f 83 Strateg Use conseration of oentu Let e = electron, neutrino, and n = nucleus Solution (a) Find the direction of otion of the recoiling daughter nucleus pe p pn 0, so pn pe p pe 0 pe and pn pe p 0 p p ( p 0) Find the angle with respect to the electron s direction 19 1 p tan tan CCW fro the electron s direction p e (b) Find the oentu of the recoiling daughter nucleus p p p ( p ) ( p ) ( 8010 kg s) (50010 kg s) n n n e kg s Thus, 19 p n kg /s in the direction found in (a) 84 Strateg Use conseration of oentu and the definition of center of ass Let the pier be to the left of the raft and woan at 0 Solution (a) Since p 0, as the woan walks toward the pier, the raft oes awa fro the pier, and the center of wwi rri wwf rrf ass does not change So, w r w r Initiall, is to the right of ri When the woan has walked to the other end of the raft, is to the left of rf B setr, the distance ri equals the distance rf, thus rf ri, so rf ri The final distance of the raft fro the dock, d f, is equal to the difference between rf and half its length,

24 Chapter 7: Linear Moentu Phsics df rf 30 ri 30 ( ) Calculate (600 kg)(65 ) (10 kg)(35 ) 45, so d f (45 ) kg 10 kg (b) Find the distance the woan walked relatie to the pier w wf wi df wi Strateg We ust deterine the initial speeds of the two cars The collision is perfectl inelastic, so the final elocities of the cars are identical Use conseration of oentu and the work-kinetic energ theore Solution Let the 1100-kg car be (1) and the 1300-kg car be () Use the work-kinetic energ theore to deterine the kinetic energ and, thus, the initial speed of the wrecked cars, which is the final speed of the collision f N i W Fr f kr kgr K 0 i, so i kg r i Thus, the final speed of the collision is f kg r Find the initial speeds pi 11i i 11i 0 pf ( 1) f, so kg 1 k h 1i f k gr cos150 (080)(980 s )(17 ) cos kg 0778 s 110 k h pi 11i i 0 i pf ( 1) f, so kg 1 k h i f k grsin150 (080)(980 s )(17 ) sin kg 0778 s 54 k h Since 110 > 70, the lighter car was speeding 86 Strateg Use the ipulse-oentu theore Solution Find the speed of the epelled gas relatie to the ground 4 p ( ) gas Fa 60 F 10 N a gas, so gas 740 s t t t t 81 kg s 87 Strateg Use the definition of linear oentu and the ipulse-oentu theore Solution Find the force the kinesin olecule needs to delier in order to accelerate the organelle 15 6 p (00110 kg)(110 s0) 18 Fa 10 N t t s 88 Strateg Use conseration of oentu The collision is perfectl inelastic, so Af Bf f 394

25 Phsics Chapter 7: Linear Moentu Solution Find the final speed in ters of the initial speed AAf BBf ( A B) f faaibbiai0 i, so f i 3 Calculate the ratio of the final kinetic energ to the initial kinetic energ i i 1 K ( A f B) f K i 3 Ai i i 89 Strateg Use conseration of energ and oentu Let B A Solution Find the aiu kinetic energ of A alone and, thus, its speed just before it strikes B 1 K 1 0 U gh 0, so 1 gh Use conseration of oentu to find the speed of the cobined bobs just after ipact The collision is perfectl inelastic, so Af Bf 1 AAfBBf( ) AAiBBi 10, so 1 3 Find the aiu height K 0 gh U 0 gh, so h h Strateg The center of ass of the disk prior to drilling is (, ) (0, 0) Let S stand for the sall circle reoed and L stand for the large circle that reains Solution Find the center of ass of the etal disk after the hole has been drilled LL SS 0, so L S S AS rs (15 c) L S S S ( 15 c) 050 c A r r (30 c) (15 c) B setr, L L L S L S 0, so ( L, L ) (050 c, 0) 91 Strateg Use conseration of oentu and energ The collision is elastic, so kinetic energ is consered Solution Find the speed of bob B iediatel after the collision Moentu conseration: Af Bf Ai Bi Ai 0, so Bf Ai Af Perfectl elastic collision ( Ki Kf): 1 Af 1 Bf Ai Bi Ai 0, so Af Bf Ai Energ conseration: 1 Ai gh, so Ai gh Find Af in ters of Ai 395

26 Chapter 7: Linear Moentu Phsics Ai Af Bf Af ( Ai Af ) Af Ai AiAf Af, so Af ( Af Ai ) 0 Thus, Af 0 or Ai The onl wa Af could equal Ai is if bob B didn t eist, so Af 0 Calculate Bf Bf Ai Af gh 0 (980 s )(51 ) 10 s 9 Strateg Use conseration of oentu and energ The collision is elastic, so kinetic energ is consered Let the positie direction be to the right Solution Find the elocities of the gliders after the collision Moentu conseration: 1f f 1i i 1i 0, so f 1i 1f Perfectl elastic collision due to bupers ( Ki Kf): 1 1f 1 f i i 1i 0, so 1f f 1i in ters of 1i Find 1f 1f 1i 1f 1f 1i 1i 1f 1f 1i 1f 1f 1i ( ), so ( ) 0 So, 1f 0 or 1i The onl wa 1f could equal 1i is if glider didn t eist, so 1f 0 Calculate f f 1i 1f 00 s 0 00 s After the collision, glider 1 is stationar and glider has a elocit of 00 s in the direction of glider 1 s initial elocit 93 Strateg Use conseration of oentu and Eq (6-6) for the kinetic energies Since the radiu nucleus is at rest, pi pra 0 Solution (a) Find the ratio of the speed of the alpha particle to the speed of the radon nucleus pf RnRn pi 0, so RnRn Therefore, Rn Rn u 111, 4 u 4 where the negatie was dropped because speed is nonnegatie 396

27 Phsics Chapter 7: Linear Moentu (b) Since the initial oentu is zero, Rn ; therefore, p p p p p p 1 (c) Find the ratio of the kinetic energies 1 1 Rn Rn Rn Rn Rn K 4 u K u Rn Rn 397