Chapter 9 Problems PDF

Problems 243 be elastic: some kinetic energy will be converted to other at the center of mass of the triangle. Draw a fourth line perpen- forms. With the assumption of some k...

Problems 243 be elastic: some kinetic energy will be converted to other at the center of mass of the triangle. Draw a fourth line perpen- forms. With the assumption of some kinetic energy lost in dicular to the longer non-hypotenuse leg, passing through the an inelastic collision, will the required speed of the eagle be center of mass, and ending as it crosses the hypotenuse of the higher or lower than in part (a)? triangle. Punch a hole in the cardboard just inside the edge of the triangle where the fourth line crosses the hypotenuse. 2. Activity Carefully draw a right triangle on a piece of card- Carefully cut the triangle out of the cardboard. Tie a string board, such that one of its non-hypotenuse legs is 30–40 cm through the hole and hang the triangle from the string. The in length and the other leg is much shorter. Measure the exact longer side of the triangle should be parallel to the table. Why midpoint of each of the three sides of the triangle and mark should this be true? Now measure the distance along the lon- these three points. Draw a line across the triangle, from a cor- ger leg from the smaller angle to the fourth line. What fraction ner of the triangle to the midpoint of the opposite side. Repeat of the entire longer leg is this distance? for the other two corners. The three lines happen to intersect Problems See the Preface for an explanation of the icons used in this problems set. m 5 0.350 kg. (c) Is the original energy in the spring or For additional assessment items for this section, go to in the cord? (d) Explain your answer to part (c). (e) Is the momentum of the system conserved in the bursting-apart process? Explain how that is possible considering (f) there Section 9.1 Linear Momentum are large forces acting and (g) there is no motion before- 1. A particle of mass m moves with momentum of magnitude p. hand and plenty of motion afterward? (a) Show that the kinetic energy of the particle is K 5 p 2/2m. 6. When you jump straight up as high as you can, what is the (b) Express the magnitude of the particle’s momentum in order of magnitude of the maximum recoil speed that you terms of its kinetic energy and mass. give to the Earth? Model the Earth as a perfectly solid object. ⁄ ⁄ 2. A 3.00-kg particle has a velocity of s3.00 i 2 4.00 jd m/s. (a) In your solution, state the physical quantities you take as Find its x and y components of momentum. (b) Find the data and the values you measure or estimate for them. magnitude and direction of its momentum. 3. A baseball approaches home plate at a speed of 45.0 m/s, Section 9.3 Analysis Model: Nonisolated System (Momentum) moving horizontally just before being hit by a bat. The batter 7. A glider of mass m is free to slide along a horizontal air hits a pop-up such that after hitting the bat, the baseball is track. It is pushed against a launcher at one end of the moving at 55.0 m/s straight up. The ball has a mass of 145 g track. Model the launcher as a light spring of force constant and is in contact with the bat for 2.00 ms. What is the average k compressed by a distance x. The glider is released from vector force the ball exerts on the bat during their interaction? rest. (a) Show that the glider attains a speed of v 5 x(k/m)1/2. (b) Show that the magnitude of the impulse imparted to the glider is given by the expression I 5 x(km)1/2. (c) Is more Section 9.2 Analysis Model: Isolated System (Momentum) work done on a cart with a large or a small mass? 4. A 65.0-kg boy and his 40.0-kg sister, both wearing roller 8. You and your brother argue often about how to safely secure blades, face each other at rest. The girl pushes the boy hard, CR a toddler in a moving car. You insist that special toddler sending him backward with velocity 2.90 m/s toward the seats are critical in improving the chances of a toddler sur- west. Ignore friction. (a) Describe the subsequent motion of viving a crash. Your brother claims that, as long as his wife the girl. (b) How much potential energy in the girl’s body is is buckled in next to him with a seat belt while he drives, she converted into mechanical energy of the boy–girl system? can hold onto their toddler on her lap in a crash. You decide (c) Is the momentum of the boy–girl system conserved in to perform a calculation to try to convince your brother. the pushing-apart process? If so, explain how that is possible Consider a hypothetical collision in which the 12-kg tod- considering (d) there are large forces acting and (e) there is dler and his parents are riding in a car traveling at 60 mi/h no motion beforehand and plenty of motion afterward. relative to the ground. The car strikes a wall, tree, or another 5. Two blocks of masses m and 3m car, and is brought to rest in 0.10 s. You wish to demonstrate are placed on a frictionless, hor- 3m to your brother the magnitude of the force necessary for his m V izontal surface. A light spring wife to hold onto their child during the collision. is attached to the more mas- 9. The front 1.20 m of a 1 400-kg car is designed as a “crumple sive block, and the blocks are Before T zone” that collapses to absorb the shock of a collision. If a car pushed together with the spring a traveling 25.0 m/s stops uniformly in 1.20 m, (a) how long between them (Fig. P9.5). S v 2.00 m/s does the collision last, (b) what is the magnitude of the aver- A cord initially holding the age force on the car, and (c) what is the magnitude of the blocks together is burned; after 3m acceleration of the car? Express the acceleration as a multiple that happens, the block of mass m of the acceleration due to gravity. 3m moves to the right with a speed of 2.00 m/s. (a) What is After 10. The magnitude of the net force exerted in the x direction the velocity of the block of mass b on a 2.50-kg particle varies in time as shown in Figure P9.10 m? (b) Find the system’s original (page 244). Find (a) the impulse of the force over the 5.00-s elastic potential energy, taking Figure P9.5 time interval, (b) the final velocity the particle attains if it is Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 244 Chapter 9 Linear Momentum and Collisions originally at rest, (c) its final F (N) vertically aligned, both balls are released velocity if its original velocity 4 from rest at the same time, to fall through ⁄ is 22.00 i m/s, and (d) the 3 a distance of 1.20 m. (a) Find the mag- average force exerted on the nitude of the downward velocity with particle for the time interval 2 which the basketball reaches the ground. between 0 and 5.00 s. 1 (b) Assume that an elastic collision with 11. Water falls without splashing the ground instantaneously reverses the t (s) at a rate of 0.250 L/s from a 0 1 2 3 4 5 velocity of the basketball while the tennis Figure P9.17 height of 2.60 m into a bucket ball is still moving down. Next, the two balls meet in an elastic Figure P9.10 collision. To what height does the tennis ball rebound? of mass 0.750 kg on a scale. If the bucket is originally empty, what does the scale read in 18. (a) Three carts of masses m1 5 4.00 kg, m 2 5 10.0 kg, and newtons 3.00 s after water starts to accumulate in it? m 3 5 3.00 kg move on a frictionless, horizontal track with speeds of v 1 5 5.00 m/s to the right, v 2 5 3.00 m/s to the Section 9.4 Collisions in One Dimension right, and v 3 5 4.00 m/s to the left as shown in Figure P9.18. Velcro couplers make the carts stick together after colliding. 12. A 1 200-kg car traveling initially at v Ci 5 25.0 m/s in an east- Find the final velocity of the train of three carts. (b) What erly direction crashes into the back of a 9 000-kg truck mov- If? Does your answer in part (a) require that all the carts ing in the same direction at v Ti 5 20.0 m/s (Fig. P9.12). The collide and stick together at the same moment? What if they velocity of the car immediately after the collision is v Cf 5 collide in a different order? 18.0 m/s to the east. (a) What is the velocity of the truck immediately after the collision? (b) What is the change in v1 v2 v3 mechanical energy of the car–truck system in the collision? (c) Account for this change in mechanical energy. m1 m2 m3 S S S S vCi vTi vCf vTf Figure P9.18 Section 9.5 Collisions in Two Dimensions 19. You have been hired as an expert witness by an attorney Before After CR for a trial involving a traffic accident. The attorney’s client, the plaintiff in this case, was traveling eastbound toward an Figure P9.12 intersection at 13.0 m/s as measured just before the acci- 13. A railroad car of mass 2.50 3 104 kg is moving with a speed dent by a roadside speed meter, and as seen by a trustworthy of 4.00 m/s. It collides and couples with three other coupled witness. As the plaintiff entered the intersection, his car railroad cars, each of the same mass as the single car and mov- was struck by a northbound driver, the defendant in this ing in the same direction with an initial speed of 2.00 m/s. case, driving a car with identical mass to the plaintiff’s. The (a) What is the speed of the four cars after the collision? (b) vehicles stuck together after the collision and left parallel What is the decrease in mechanical energy in the collision? skid marks at an angle of u 5 55.08 north of east, as meas- 14. Four railroad cars, each of mass 2.50 3 104 kg, are coupled ured by accident investigators. The defendant is claiming together and coasting along horizontal tracks at speed vi that he was traveling within the 35-mi/h speed limit. What toward the south. A very strong but foolish movie actor, advice do you give to the attorney? riding on the second car, uncouples the front car and gives 20. Two shuffleboard disks of equal mass, one orange and the it a big push, increasing its speed to 4.00 m/s southward. V other yellow, are involved in an elastic, glancing collision. The remaining three cars continue moving south, now at The yellow disk is initially at rest and is struck by the orange 2.00 m/s. (a) Find the initial speed of the four cars. (b) By disk moving with a speed of 5.00 m/s. After the collision, how much did the potential energy within the body of the the orange disk moves along a direction that makes an actor change? (c) State the relationship between the process angle of 37.08 with its initial direction of motion. The veloc- described here and the process in Problem 13. ities of the two disks are perpendicular after the collision. 15. A car of mass m moving at a speed v 1 collides and couples Determine the final speed of each disk. with the back of a truck of mass 2m moving initially in the 21. Two shuffleboard disks of equal mass, one orange and the same direction as the car at a lower speed v 2. (a) What is the other yellow, are involved in an elastic, glancing collision. speed vf of the two vehicles immediately after the collision? The yellow disk is initially at rest and is struck by the orange (b) What is the change in kinetic energy of the car–truck disk moving with a speed vi. After the collision, the orange system in the collision? disk moves along a direction that makes an angle u with its 16. A 7.00-g bullet, when fired from a gun into a 1.00-kg block initial direction of motion. The velocities of the two disks of wood held in a vise, penetrates the block to a depth of are perpendicular after the collision. Determine the final 8.00 cm. This block of wood is next placed on a frictionless speed of each disk. horizontal surface, and a second 7.00-g bullet is fired from 22. A 90.0-kg fullback running east with a speed of 5.00 m/s is the gun into the block. To what depth will the bullet penet- tackled by a 95.0-kg opponent running north with a speed rate the block in this case? V of 3.00 m/s. (a) Explain why the successful tackle consti- 17. A tennis ball of mass 57.0 g is held just above a basketball tutes a perfectly inelastic collision. (b) Calculate the velocity T of mass 590 g as shown in Figure P9.17. With their centers of the players immediately after the tackle. (c) Determine Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Problems 245 the decrease in mechanical energy as a result of the colli- time, the vector position of a 5.50 g particle varies as S ⁄ ⁄ ⁄ sion. Account for this decrease. r 2 5 3 i 2 2 i t 2 2 6 jt. At t 5 2.50 s, determine (a) the vector ⁄ position of the center of mass of the system, (b) the linear 23. A proton, moving with a velocity of vi i , collides elastically momentum of the system, (c) the velocity of the center of with another proton that is initially at rest. Assuming that the mass, (d) the acceleration of the center of mass, and (e) the two protons have equal speeds after the collision, find (a) the net force exerted on the two-particle system. speed of each proton after the collision in terms of vi and (b) the direction of the velocity vectors after the collision. 29. You have been hired as an expert witness in an investiga- CR tion of a quadcopter drone incident. The incident occurred Section 9.6 The Center of Mass during a very rare meteor shower during which several unusually massive chunks of meteoric material were passing 24. A uniform piece of sheet y (cm) through the atmosphere and striking the ground. The V metal is shaped as shown in 30 unmanned drone was hovering at rest over the center of Figure P9.24. Compute the x a house on fire, having just dropped fire retardant, when and y coordinates of the cen- 20 it seemed to spontaneously explode into four large pieces. ter of mass of the piece. The locations of the four pieces on the ground were meas- 10 ured as follows, relative to the center of the house over 25. Explorers in the jungle find an ancient monument in the x (cm) which the drone was hovering: shape of a large isosceles trian- 10 20 30 Distance from Center Direction gle as shown in Figure P9.25. The monument is made from Figure P9.24 Piece # Mass (kg) of House (m) from House tens of thousands of small stone blocks of density 3 800 kg/m3. 1 80.0 150 Due west The monument is 15.7 m high and 64.8 m wide at its base 2 120 75.0 Due north and is everywhere 3.60 m thick from front to back. Before 3 50.0 90.0 20.0° west the monument was built many years ago, all the stone blocks of south lay on the ground. How much work did laborers do on the 4 150 50.0 20.0° north blocks to put them in position while building the entire mon- of east ument? Note: The gravitational potential energy of an object– Earth system is given by Ug 5 Mgy CM, where M is the total The fire department is suggesting that the drone was defec- mass of the object and y CM is the elevation of its center of tive and exploded while in use. The drone manufacturer is mass above the chosen reference level. suggesting that the drone was struck by a meteorite, causing the explosion. Perform a calculation that will show evidence suggesting agreement with one of these positions. 15.7 m Section 9.8 Deformable Systems 64.8 m 30. For a technology project, a stu- 3.60 m dent has built a vehicle, of total mass 6.00 kg, that moves itself. As shown in Figure P9.30, it runs on four light wheels. A reel Figure P9.25 is attached to one of the axles, 26. A rod of length 30.0 cm has linear density (mass per length) and a cord originally wound on given by the reel goes up over a pulley attached to the vehicle to support l 5 50.0 1 20.0x an elevated load. After the vehicle where x is the distance from one end, measured in meters, is released from rest, the load des- and l is in grams/meter. (a) What is the mass of the rod? (b) cends very slowly, unwinding the How far from the x 5 0 end is its center of mass? cord to turn the axle and make the vehicle move forward (to the left in Fig. P9.30). Friction is neg- Figure P9.30 Section 9.7 Systems of Many Particles ligible in the pulley and axle bearings. The wheels do not 27. Consider a system of two particles in the xy plane: m1 5 2.00 kg slip on the floor. The reel has been constructed with a con- ⁄ ⁄ is at the location S r 1 5 s1.00i 1 2.00j d m and has a velocity of ical shape so that the load descends at a constant low speed ⁄ ⁄ ⁄ ⁄ s3.00i 1 0.500j d m/s; m 2 5 3.00 kg is at S r 2 5 s24.00i 2 3.00j d m while the vehicle moves horizontally across the floor with ⁄ ⁄ and has velocity s3.00i 2 2.00j d m/s. (a) Plot these particles ⁄ constant acceleration, reaching a final velocity of 3.00 i m/s. on a grid or graph paper. Draw their position vectors and (a) Does the floor impart impulse to the vehicle? If so, how show their velocities. (b) Find the position of the center of much? (b) Does the floor do work on the vehicle? If so, how mass of the system and mark it on the grid. (c) Determine the much? (c) Does it make sense to say that the final momentum velocity of the center of mass and also show it on the diagram. of the vehicle came from the floor? If not, where did it come (d) What is the total linear momentum of the system? from? (d) Does it make sense to say that the final kinetic 28. The vector position of a 3.50-g particle moving in the xy energy of the vehicle came from the floor? If not, where did ⁄ ⁄ ⁄ plane varies in time according to S r 1 5 s3 i 1 3 j dt 1 2 jt 2, it come from? (e) Can we say that one particular force causes S where t is in seconds and r is in centimeters. At the same the forward acceleration of the vehicle? What does cause it? Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 246 Chapter 9 Linear Momentum and Collisions 31. A 60.0-kg person bends his knees and then jumps straight up. to safety by throwing her gloves of mass m in the direction After his feet leave the floor, his motion is unaffected by air res- opposite the safe side. (a) She throws the gloves as hard as istance and his center of mass rises by a maximum of 15.0 cm. she can, and they leave her hand with a horizontal veloc- Model the floor as completely solid and motionless. (a) Does ity S vgloves. Explain whether or not she moves. (b) If she does the floor impart impulse to the person? (b) Does the floor move, calculate her velocity S vgirl relative to the Earth after do work on the person? (c) With what momentum does the she throws the gloves. (c) Discuss her motion from the point person leave the floor? (d) Does it make sense to say that this of view of the forces acting on her. momentum came from the floor? Explain. (e) With what kin- 36. (a) Figure P9.36 shows three points in the operation of the etic energy does the person leave the floor? (f) Does it make ballistic pendulum discussed in Example 9.6 (and shown in sense to say that this energy came from the floor? Explain. Fig. 9.10b). The projectile approaches the pendulum in Section 9.9 Rocket Propulsion Figure P9.36a. Figure P9.36b shows the situation just after the projectile is captured in the pendulum. In Figure P9.36c, 32. A garden hose is held as the pendulum arm has swung upward and come to rest shown in Figure P9.32. momentarily at a height h above its initial position. Prove that The hose is originally full the ratio of the kinetic energy of the projectile–pendulum of motionless water. What system immediately after the collision to the kinetic energy additional force is neces- immediately before is m1/(m1 1 m 2). (b) What is the ratio of sary to hold the nozzle the momentum of the system immediately after the collision stationary after the water to the momentum immediately before? (c) A student believes flow is turned on if the dis- that such a large decrease in mechanical energy must be charge rate is 0.600 kg/s accompanied by at least a small decrease in momentum. How with a speed of 25.0 m/s? Figure P9.32 would you convince this student of the truth? 33. A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of 10 000 m/s. (a) It has an engine and fuel designed to produce an exhaust speed of 2 000 m/s. How much fuel plus oxidizer is required? (b) If a dif- ferent fuel and engine design could give an exhaust speed of 5 000 m/s, what amount of fuel and oxidizer would be vi vf h required for the same task? (c) Noting that the exhaust m1 m2 speed in part (b) is 2.50 times higher than that in part (a), explain why the required fuel mass is not simply smaller by a b c a factor of 2.50. 34. A rocket has total mass Mi 5 360 kg, including M fuel 5 Figure P9.36 Problems 36 and 43. (a) A metal ball moves toward the pendulum. (b) The ball is captured 330 kg of fuel and oxidizer. In interstellar space, it by the pendulum. (c) The ball–pendulum combination starts from rest at the position x 5 0, turns on its swings up through a height h before coming to rest. engine at time t 5 0, and puts out exhaust with rel- ative speed ve 5 1 500 m/s at the constant rate k 5 37. Review. A 60.0-kg person running at an initial speed of 2.50 kg/s. The fuel will last for a burn time of T b 5 M fuel/k 5 4.00 m/s jumps onto a 120-kg cart initially at rest (Fig. P9.37). 330 kg/(2.5 kg/s) 5 132 s. (a) Show that during the burn The person slides on the cart’s top surface and finally comes the velocity of the rocket as a function of time is given by to rest relative to the cart. The coefficient of kinetic friction vstd 5 2ve ln 1 2 S kt Mi D between the person and the cart is 0.400. Friction between the cart and ground can be ignored. (a) Find the final velo- city of the person and cart relative to the ground. (b) Find the (b) Make a graph of the velocity of the rocket as a function friction force acting on the person while he is sliding across of time for times running from 0 to 132 s. (c) Show that the the top surface of the cart. (c) How long does the friction acceleration of the rocket is force act on the person? (d) Find the change in momentum of the person and the change in momentum of the cart. kve a std 5 (e) Determine the displacement of the person relative to Mi 2 kt the ground while he is sliding on the cart. (f) Determine (d) Graph the acceleration as a function of time. (e) Show the displacement of the cart relative to the ground while the that the position of the rocket is person is sliding. (g) Find the change in kinetic energy of x std 5 ve S D S Mi k 2 t ln 1 2 kt Mi D 1 ve t 60.0 kg 4.00 m/s (f) Graph the position during the burn as a function of time. Additional Problems 120 kg 35. An amateur skater of mass M is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return Figure P9.37 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Problems 247 the person. (h) Find the change in kinetic energy of the the speed v at maximum compression, (b) the maximum cart. (i) Explain why the answers to (g) and (h) differ. compression x max, and (c) the velocity of each glider after m1 (What kind of collision is this one, and what accounts for has lost contact with the spring. the loss of mechanical energy?) 42. Pursued by ferocious wolves, you are in a sleigh with no 38. A cannon is rigidly horses, gliding without friction across an ice-covered lake. attached to a carriage, You take an action described by the equations which can move along 45.0° ⁄ ⁄ ⁄ horizontal rails but is s270 kgds7.50 mysd i 5 s15.0 kgds2v1f i d 1 s255 kgdsv2f i d connected to a post v1f 1 v2f 5 8.00 mys by a large spring, ini- (a) Complete the statement of the problem, giving the data tially unstretched and and identifying the unknowns. (b) Find the values of v 1f with force constant and v 2f. (c) Find the amount of energy that has been trans- k 5 2.00 3 104 N/m, formed from potential energy stored in your body to kinetic as shown in Fig­ ure energy of the system. P9.38. The cannon Figure P9.38 fires a 200-kg projectile at a velocity of 125 m/s directed 43. Review. A student performs a ballistic pendulum experi- 45.0° above the horizontal. (a) Assuming that the mass of ment using an apparatus similar to that discussed in the cannon and its carriage is 5 000 kg, find the recoil speed Example 9.6 and shown in Figure P9.36. She obtains the fol- of the cannon. (b) Determine the maximum extension of lowing average data: h 5 8.68 cm, projectile mass m1 5 68.8 g, the spring. (c) Find the maximum force the spring exerts and pendulum mass m 2 5 263 g. (a) Determine the initial on the carriage. (d) Consider the system consisting of the speed v1A of the projectile. (b) The second part of her experi- cannon, carriage, and projectile. Is the momentum of this ment is to obtain v1A by firing the same projectile horizont- system conserved during the firing? Why or why not? ally (with the pendulum removed from the path) and meas- uring its final horizontal position x and distance of fall y 39. A 1.25-kg wooden block (Fig. P9.43). What numerical value does she obtain for v1A rests on a table over a based on her measured values of x 5 257 cm and y 5 85.3 cm? large hole as in Figure M (c) What factors might account for the difference in this P9.39. A 5.00-g bullet value compared with that obtained in part (a)? with an initial velocity vi S is fired upward into the v1A S vi bottom of the block and m remains in the block after the collision. The Figure P9.39 y block and bullet rise to Problems 39 and 40. a maximum height of 22.0 cm. (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Calculate the initial velocity of the bullet from the informa- x tion provided. Figure P9.43 40. A wooden block of mass M rests on a table over a large 44. Why is the following situation impossible? An astronaut, together hole as in Figure P9.39. A bullet of mass m with an initial with the equipment he carries, has a mass of 150 kg. He is velocity of vi is fired upward into the bottom of the block taking a space walk outside his spacecraft, which is drift- and remains in the block after the collision. The block and ing through space with a constant velocity. The astronaut bullet rise to a maximum height of h. (a) Describe how you accidentally pushes against the spacecraft and begins mov- would find the initial velocity of the bullet using ideas you ing away at 20.0 m/s, relative to the spacecraft, without a have learned in this chapter. (b) Find an expression for the tether. To return, he takes equipment off his space suit and initial velocity of the bullet. throws it in the direction away from the spacecraft. Because 41. Two gliders are set in motion on a horizontal air track. A of his bulky space suit, he can throw equipment at a max- light spring of force constant k is attached to the back end of imum speed of 5.00 m/s relative to himself. After throwing the second glider. As shown in Figure P9.41, the first glider, enough equipment, he starts moving back to the spacecraft of mass m1, moves to the right with speed v 1, and the second and is able to grab onto it and climb inside. glider, of mass m 2, moves more slowly to the right with speed 45. Review. A bullet of mass m 5 8.00 g is fired into a block of v 2. When m1 collides with the spring attached to m 2, the T mass M 5 250 g that is initially at rest at the edge of a fric- spring compresses by a distance x max, and the gliders then tionless table of height h 5 1.00 m (Fig. P9.45). The bullet move apart again. In terms of v 1, v 2, m1, m 2, and k, find (a) m S S M v1 v2 k m1 m2 h d Figure P9.45 Problems 45 and 46. Figure P9.41 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 248 Chapter 9 Linear Momentum and Collisions remains in the block, and after the impact the block lands on the particle causes its change in kinetic energy; and the d 5 2.00 m from the bottom of the table. Determine the ini- impulse–momentum theorem, stating that the total impulse tial speed of the bullet. on the particle causes its change in momentum. In this problem, you compare predictions of the three theories in 46. Review. A bullet of mass m is fired into a block of mass M ⁄ one particular case. A 3.00-kg object has velocity 7.00 j m/s. initially at rest at the edge of a frictionless table of height ⁄ Then, a constant net force 12.0 i N acts on the object for h (Fig. P9.45). The bullet remains in the block, and after 5.00 s. (a) Calculate the object’s final velocity, using the impact the block lands a distance d from the bottom of the impulse–momentum theorem. (b) Calculate its accelera- table. Determine the initial speed of the bullet. tion from S a 5 sS vf 2 S v i dyDt. (c) Calculate its acceleration S 47. A 0.500-kg sphere moving with a velocity expressed as from a 5 o F ym. (d) Find the object’s vector displacement S ⁄ ⁄ ⁄ s2.00 i 2 3.00 j 1 1.00kd m/s strikes a second, lighter from DS r 5Sv t 1 12S a t 2. (e) Find the work done on the object Si sphere of mass 1.50 kg moving with an initial velocity of from W 5 F ? DS r. (f) Find the final kinetic energy from ⁄ ⁄ ⁄ 1 1 S S s21.00 i 1 2.00 j 2 3.00kd m/s. (a) The velocity of the 0.500- 2 2 mvf 5 2 m vf ? vf. (g) Find the final kinetic energy from ⁄ ⁄ ⁄ kg sphere after the collision is s21.00 i 1 3.00 j 2 8.00kd 1 2 2 mvi 1 W. (h) State the result of comparing the answers to m/s. Find the final velocity of the 1.50-kg sphere and identify parts (b) and (c), and the answers to parts (f) and (g). the kind of collision (elastic, inelastic, or perfectly inelastic). (b) Now assume the velocity of the 0.500-kg sphere after the Challenge Problems ⁄ ⁄ ⁄ collision is (20.250 i 1 0.750 j 2 2.00k) m/s. Find the final 52. Sand from a stationary hopper falls onto a moving con- velocity of the 1.50-kg sphere and identify the kind of colli- veyor belt at the rate of 5.00 kg/s as shown in Figure P9.52. sion. (c) What If? Take the velocity of the 0.500-kg sphere ⁄ ⁄ ⁄ The conveyor belt is supported by frictionless rollers and after the collision as s21.00 i 1 3.00 j 1 a kd m/s. Find the moves at a constant speed of v 5 0.750 m/s under the value of a and the velocity of the 1.50-kg sphere after an S action of a constant horizontal external force F ext supplied elastic collision. by the motor that drives the belt. Find (a) the sand’s rate of 48. Review. A metal cannonball of mass m rests next to a tree change of momentum in the horizontal direction, (b) the at the very edge of a cliff 36.0 m above the surface of the force of friction exerted by the belt on the sand, (c) the S S ocean. In an effort to knock the cannonball off the cliff, external force F ext , (d) the work done by F ext in 1 s, and some children tie one end of a rope around a stone of mass (e) the kinetic energy acquired by the falling sand each 80.0 kg and the other end to a tree limb just above the can- second due to the change in its horizontal motion. nonball. They tighten the rope so that the stone just clears (f) Why are the answers to parts (d) and (e) different? the ground and hangs next to the cannonball. The children manage to swing the stone back until it is held at rest 1.80 m above the ground. The children release the stone, which then swings down and makes a head-on, elastic collision with the cannonball, projecting it horizontally off the cliff. The cannonball lands in the ocean a horizontal distance R v S away from its initial position. (a) Find the horizontal com- Fext ponent R of the cannonball’s displacement as it depends on m. (b) What is the maximum possible value for R, and (c) to what value of m does it correspond? (d) For the stone– Figure P9.52 cannonball–Earth system, is mechanical energy conserved 53. Two particles with masses m and 3m are moving toward throughout the process? Is this principle sufficient to solve each other along the x axis with the same initial speeds the entire problem? Explain. (e) What if? Show that R does vi. Particle m is traveling to the left, and particle 3m is not depend on the value of the gravitational acceleration. Is traveling to the right. They undergo an elastic glancing this result remarkable? State how one might make sense of it. collision such that particle m is moving in the negative 49. Review. A light spring of force constant 3.85 N/m is com- y direction after the collision at a right angle from its initial pressed by 8.00 cm and held between a 0.250-kg block on the direction. (a) Find the final speeds of the two particles in left and a 0.500-kg block on the right. Both blocks are at rest on terms of vi. (b) What is the angle u at which the particle 3m a horizontal surface. The blocks are released simultaneously so is scattered? that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction 54. On a horizontal air track, a glider of mass m carries a between each block and the surface is (a) 0, (b) 0.100, and G-shaped post. The post supports a small dense sphere, (c) 0.462. Assume the coefficient of static friction is greater also of mass m, hanging just above the top of the glider than the coefficient of kinetic friction in every case. on a cord of length L. The glider and sphere are initially at rest with the cord vertical. A constant horizontal force 50. Consider as a system the Sun with the Earth in a circular of magnitude F is applied to the glider, moving it through orbit around it. Find the magnitude of the change in the displacement x 1; then the force is removed. During the velocity of the Sun relative to the center of mass of the sys- time interval when the force is applied, the sphere moves tem over a six-month period. Ignore the influence of other through a displacement with horizontal component x 2. celestial objects. You may obtain the necessary astronomical (a) Find the horizontal component of the velocity of the data from the endpapers of the book. center of mass of the glider–sphere system when the force 51. Review. There are (one can say) three coequal theories of is removed. (b) After the force is removed, the glider con- motion for a single particle: Newton’s second law, stating tinues to move on the track and the sphere swings back that the total force on the particle causes its acceleration; and forth, both without friction. Find an expression for the work–kinetic energy theorem, stating that the total work the largest angle the cord makes with the vertical. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.