Chs 9 & 10 - we can infer the speed of...

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Unformatted text preview: we can infer the speed of ' straightforward measure- gas. Taking nitrogen at oom temperature, for ex— . more careful calculation erimental evidence (Chap- given temperature have a _.)ur formula should be re- §quared speed, 05,. And 1 of our own calculation, %——although the rigorous ble treatment of the prob- :es of having molecules I directions; it is only an esult in detail. (It would, es striking AA, instead of e simple analysis that we useful beginning for the iicroscopic terms. And, :5 is not our present con- )inted out that any con- in physics is provisional, ‘C, it is finally vindicated, ned. The most dramatic dynamics took place in elusive, neutral particle :a decay. .The prediction rent nonconservation of ‘haps the most beautiful Jrnished by the apparent PROBLEMS 357 Fig. 9—25 Evidence for the neutrino. The visible tracks of the electron and the recoiling lithium 6 nucleus in the beta decay ofhelinm 6 in a cloud- c/mmber are not collinear. [Front J. Csikai and A. Szalay, Soviet Physics JETP, 8, 749 (1959).] The situation can be simply statedas follows: It is known that the process of beta decay involves the ejection of an electron from a nucleus, as a result of which the nuclear charge goes up by one unit (if the electron is an ordinary negative electron). If no other? particles were involved, the process could be written A;->B+e‘ where A is the initial nucleus and B the final nucleus. If A were effectively isolated, and initially stationary, our belief in linear momentum conservation would lead us to predict that, whatever the direction (or energy) of the ejected electron, the nucleus B would inevitably recoil in the opposite direction. Any departure from this, regardless of all other details, would demand the involvement of another particle. Figure 9—25 shows a cloud-chamber photograph of the beta decay of helium 6. The decay takes place at the position of the sharp knee near the top of the picture. The short stubby track pointing in a “northwesterly” direction is the recoiling nucleus of lithium 6; the other track is the electron. There must be another particle—the neutrino—if the final momentum vectors are to add up to have the same resultant——i.e., zero—as the initially stationary “He nucleus. It fails to reveal itself because its lack of charge, or of almost any other interaction, allows it to escape unnoticed—so readily, in fact, that the chance would be only about 1 in 1012 of its interacting with any matter in passing right through the earth. 9—1 A particle of mass in, traveling with velocity Do, makes a com- pletely inelastic collision with an initially stationary particle of mass M. Make a graph of the final velocity u as a function of the ratio m/M from m/M = 0 to m/M = 10. 9—2 Consider how conservation of linear momentum applies to a ball bouncing off a wall. 9—3 A mouse is put into a small closed box that is placed upon a table. Could a clever mouse control the movements of the box over the table? Just what maneuvers could the mouse make the box per- form? If you were such a mouse, and your object were to elude pursuers, would you prefer that the table have a large, small, or negligible coefficient of friction? 9—4 In the Principia, Newton mentions that in one set of collision experiments he found that the relative velocity of separation of two objects of a certain kind of material Was five ninths of their relative velocity of approach. Suppose that an initially stationary object, of mass 3mg, of this material was struck by a similar object of mass 2mo, _- traveling with an initial velocity 00. Find the final velocities of both objects. ' 9—5 A particle of mass mo, traveling at speed (:0, strikes a stationary particle of mass 2mg. As a result, the particle of mass mo is deflected through 45° and has a final speed of 00/2. Find the speed and direc- tion of the particle of mass 2mg after this collision. Was kinetic energy conserved ? 9—6 Two skaters (A and B), both of mass 70 kg, are approaching one another, each with a speed of 1 m/sec. A carries a bowling ball with a mass of 10 kg. Both skaters can toss the ball at 5 m/sec relative to themselves. To avoid collision they start tossing the ball back and forth when they are 10 m apart. Is one toss enough? How about two tosses, i.e., A gets the ball back? If the ball weighs half as much but they can throw twice as fast, how many tosses do they need? Plot the entire incident on a time versus displacement graph, in which the positions of the skaters are marked along the abscissa, and the advance of time is represented by the increasing value of the ordinate. (Mark the initial positions of the skaters at x i5 m, and include the space—time record of the ball’s motion in the diagram.) This situation serves as a simple model of the present view of interactions (repulsive, in the above example) between elementary particles. An attractive interaction can be simulated by supposing that the skaters exchange a boomerang instead of a ball. [These theoretical models were pre- sented by F. Reines and J. P. F. Sellschop in an article entitled “Neutrinos from the Atmosphere and Beyond," Sci. Am., 214, 40 (Feb. 1966).] ll 9—7 Find the average recoil force on a machine gun firing 240 rounds (shots) per minute, if the mass of each bullet is 10g and the muzzle velocity is 900 m/sec. ty 0 as a function of the ratio inear momentum applies to a sed box that is placed upon a he movements of the box over the mouse make the box per- nd your object were to elude table have a large, small, or ns that in one set of collision velocity of separation of two as five ninths of their relative it initially stationary object, of y a similar object of mass 2mo, 'ind the final velocities of both ll. speed 00, strikes a stationary particle of mass mu is deflected )/2. Find the speed and direc- is collision. Was kinetic energy T mass 70 kg, are approaching /sec. A carries a bowling ball toss the ball at 5 m/sec relative start tossing the ball back and nne toss enough? How about If the ball weighs half as much iany tosses do they need? Plot placement graph, in which the g the abscissa, and the advance value of the ordinate. (Mark x = i5 m, and include the n the diagram.) This situation view of interactions (repulsive, Jtary particles. An attractive sing that the skaters exchange 3 theoretical models were pre- :llschop in an article entitled l Beyond,” Sci. Am., 214, 40 machine gun firing 240 rounds bullet is 10 g and the muzzle 359 9—8 Water emerges in a vertical jet from a nozzle mounted on one end of a long horizontal metal tube, clamped at its other end and thin enough to be rather flexible. The jet rises to a height of 2.5 m above the nozzle, and the rate of water flow is 21iter/min. It has been previously found by static experiments that the nozzle is de— pressed vertically by an amount proportional to the applied force, and that a mass of 10g, hung upon it, causes a depression of 1 cm. How far is the nozzle depressed by the reaction force from the water jet ? {This problem is based on a demonstration experiment described by E. F. Schrader, Am, J. Phys., 33, 784 (1965).] 9—9 A “standard fire stream" employed by a city fire department delivers 250 gallons of water per minute and can attain a height of 70 ft on a building whose base is 63 ft fromrthe nozzle. Neglecting air resistance: (a) What is the nozzle velocity of the stream? (b) If directed horizontally against a vertical wall, what force would the stream exert? (Assume that the water spreads out over the surface of the wall without any rebound, so that the collision is ef- fectively inelastic.) 9—10 A helicopter has a total mass M. Its main rotor blade sweeps out a circle of radius R, and air over this whole circular area is pulled in from above the rotor and driven vertically downward with a speed 00. The density of air is p. (a) If the helicopter hovers at some fixed height, what must be the value of vi)? (b) One of the largest helicopters of the type described above weighs about 10 tons and has R ’2 10 m. What is no for hovering in this case? Take p = 1.3 kg/ma. 9—11 A rocket of initial mass Mo ejects its burnt fuel at a constant rate IdM/dt] = ,u and at a speed 1:0 relative to the rocket. (a) Calculate the initial acceleration of the rocket if it starts vertically upward from its launch pad. (b) If 1:0 = 2000 m/sec, how many kilograms of fuel must be ejected per second to give such a rocket, of mass 1000 tons, an initial upward acceleration equal to 0.5 g? 9—12 This rather complicated problem is designed to illustrate the advantage that can be obtained by the use of multiple-stage instead of single—stage rockets as launching vehicles. Suppose that the pay- load (e.g.,'a space capsule) has mass m and is mounted on a two-stage rocket (see the figure). The total mass—~both rockets fully fueled, plus . the payload—is Nm. The mass of the second-stage rocket plus the payload, after first-stage burnout and separation, is nm. In each stage the ratio of burnout mass (casing) to initial mass (casing plus fuel) is r, and the exhaust speed is 00. (a) Show that the velocity 1:; gained from first-stage burn, i." i sgtfitfi REE}; W I I nm \———————v———————/ Nm l starting from rest (and ignoring gravity), is given by _ ,[_N_] ”"”0" rN+n(1——r) (b) Obtain a corresponding expression for the additional velocity, 02, gained from the second-stage burn. (c) Adding v1 and 02, you have the payload velocity v in terms of N, n, and r. Taking N and r as constants, find the value of n for which 0 is a maximum. (d) Show that the condition for u to be a maximum corresponds to having equal gains of velocity in the two stages. Find the maximum value of v, and verify that it makes sense for the limiting cases de- scribed by r = 0 and r = l. (e) Find an expression for the payload velocity of a single-stage rocket with the same values of N, r, and 00. (f) Suppose that it is desired to obtain a payload velocity of 10 km/sec, using rockets for which 120 = 2.5 km/sec and r = 0.1. Show that the job can be done with a two-stage rocket but is im— possible, however large the value of N, with a single-stage rocket. (g) If you are ambitious, try extending the analysis to an arbi- trary number of stages. It is possible to show that once again the greatest payload velocity for a given total initial mass is obtained if the stages are so designed that the velocity increment contributed by each stage is the same. 9—13 A block of mass m, initially at rest on a frictionless surface, is bombarded by a succession of particles each of mass 5m (<< m) and of initial speed we in the positive x direction. The collisions are per- fectly elastic and each particle bounces back in the negative x direction. Show that the speed acquired by the- block after the nth particle has struck it is given very nearly by u = you -- e’m‘), wherea = 26m/m. Consider the validity of this result for an << 1 as well as for an ——+ 00. 9—14 Newton calculated the resistive force for an object traveling through a fluid by supposing that the particles of the fluid (supposedly initially stationary) rebounded elastically when struck by the object. (a) On this model, the resistive force would vary as some power, n, of the speed 17 of the object. What is the value of n? ' (b) Suppose that a flat—ended object of cross-sectional area A is moving at speed 12 through a fluid of density p. By picturing the fluid e additional velocity, 1 velocity v in terms d the value of n for .ximum corresponds Find the maximum e limiting cases de- ity of a single—stage payload velocity of l/SCC and r = 0.1. : rocket but is im- e—stage rocket. tnalysis to an arbi- iat once again the ss is obtained if the ontributed by each tionless surface, is ass 6m (<< m) and collisions are per- :gative x direction. re nth particle has Iherea = 26m/m. ll as for an ——> oo . I object traveling 'fluid (supposedly k by the object. ry as some power, It? ‘ ' ectional area A is )icturing the fluid as composed of n particles, each of mass m, per unit volume (such that nm = p), obtain an explicit expression for the resistive force if .eachrparticle that is struck by the object recoils elastically from it. (c) If the object, instead of being flat—ended, were a massive sphere of radius r, traveling at speed 0 through a medium of density p, what would the magnitude of the resistive force be? The whole cal- culation can be carried out from the standpoint of a frame attached to the sphere, so that the fluid particles approach it with the velocity —v. Assume that in this frame the fluid particles are reflected as by a mirror—angle of reflection equals angle of incidence (see the figure). You must consider the surface of the sphere as divided up into circular zones corresponding to small angular increments d6 at the various possible values of 6. 9—15 A particle of mass mi and initial velocity Lu strikes a stationary particle of mass mg. The collision is perfectly elastic. It is observed that after the collision the particles have equal and opposite velocities. Find (a) The ratio mg/ml. (b) The velocity of the center of mass. (c) The total kinetic energy of the two particles in the center of mass frame expressed as a fraction of @mufi’. (d) The final kinetic energy of mi in the lab frame. 9~l6 A mass ml collides with a mass 1112. Define relative velocity as the velocity of mi observed in the rest frame of mg. Show the equivalence of the following two statements: (1) Total kinetic energy is conserved. (2) The magnitude of the relative velocity is unchanged. (It is suggested that you solve the problem for a one-dimensional collision, at least in the first instance.) 9—17 A collision apparatus is made of a set of n graded masses sus— pended so that they are in a horizontal line and not quite in contact with one another (see the figure). The first mass is firm, the second is fgmu, the third Fm“, and so on, so that the last mass is _/'"nm. The first mass is struck by a particle of mass mo traveling at a speed uo. This produces a succession of collisions along the line of masses. U0 O ’"u .fi’ln f"m" (a) Assuming that all the collisions are perfectly elastic, show that the last mass flies off with a speed 0,. given by u ~ (—2—); 'n 1 + j 0 (b) Hence show that, iffis close to unity, so that it can be written as l :l: c (with e<< 1), this system can be used to transfer virtually all the kinetic energy of the incident mass to the last one, even for large n. i (c) For f = 0.9, n = 20, calculate the mass, speed, and kinetic energy of the last mass in the line in terms of the mass, speed, and kinetic energy of the incident particle. Compare this with the result of a direct collision between the incident mass and the last mass in the line. 9—18 A 2—kg and an 8-kg mass collide elastically, compressing a spring bumper on one of them; the bumper returns to its original length as the masses separate. Assume that the collision takes place along a single line and that you can cause the collision to occur in diflercnt ways, each having the same initial energy: Case A: The 8-kg mass has 16.! of kinetic energy and hits the sta- tionary 2—kg mass. Case B: The 2—kg mass has 16.! of kinetic energy and hits the sta- tionary 8-kg mass. (at) Which way of causing the collision to occur will result in the greater compression of the spring? Arrive at your choice without actually solving for the compression of the spring. ' (b) Keeping the condition of a total initial kinetic energy of 16], how should this energy be divided between the two masses to obtain the greatest possible compression ol~ the spring? 9—»1‘) In a certain road accident (this is based on an aetual case) a ear of mass 2000kg, traveling south, collided in the middle ol~ an inter- section* with a truck of mass 6000kg, traveling west. The vehicles locked and skidded oil the road along a line pointing almost exactly southwest. A witness claimed that the truck had entered the inter— section at 50 mph. (a) Do you believe the witness? (b) Whether or not you believe him, what fraction ol‘ the total ./"‘mn ie collisions are perfectly elastic, show a speed Cu given by fis close to unity, so that it can be ), this system can be used to transfer of the incident mass to the last one, calculate the mass, speed, and kinetic line in terms of the mass, speed, and )article. Compare this with the result 1e incident mass and the last mass in 188 collide elastically, compressing a 1; the bumper returns to its original Assume that the collision takes place -u can cause the collision to occur in same initial energy: J of kinetic energy and hits the sta- J of kinetic energy and hits the sta- ; the collision to occur will result in pring'? Arrive at your choice withom sion of the spring. ‘ i of a total initial kinetic energy of : divided between the two masses to aression of the spring '.’ (this is based on an actual case) a car i, collided in the middle of an inter— )00 kg, traveling west. The vehicles along a line pointing almost exactly hat the truck had entered the inter— less ‘3 'lievc him, what fraction of the total 363 initial kinetic energy was converted into other forms of energy by the collision? 9—20 A nucleus A of mass 2m, traveling with a velocity u, collides with a stationary nucleus of mass 10m. The collision results in a change of the total kinetic energy. After collision the nucleus A is observed to be traveling with speed 01 at 90° to its original direction of motion, and B is traveling with speed 122 at angle 6 (sin 6 = 3/5) to the original direction of motion of A. (a) What are the magnitudes of 1:1 and 02? (b) What fraction of the initial kinetic energy is gained or lost as a result of the interaction? 9—21 A particle of mass m and initial velocityu collides elastically with a particle of mass M initially at rest. As a result of the_collision the particle of mass m is deflected through 90° and its Speed is reduced to u/V/i. The particle of mass M recoils with speed v at an angle 6 to the original direction of m. (All speeds and angles are those observed in the laboratory system.) (a) Find M in terms of m, and e in terms of 1:. Find also the angle 0. (b) At what angles are the particles deflected in the center-of- mass system? 9—22 Make measurements on the stroboscopic photographs of a col- lision of two magnetized pucks (Fig. 9’23) to test the conservation of linear momentum and total kinetic energy between the initial state (first three time units) and the final state (last three time units). 9-23 A particle of mass 2m and of velocity u strikes a second particle of mass 2m initially at rest. As a result of the collision, a particle of mass m is produced which moves ofi‘ at 45° with respect to the initial direction of the incident particle. The other product of this rearrange- ment collision is a particle of mass 3m. Assuming that this collision involves no significant change of total kinetic energy, calculate the speed and direction of the particle of mass 3m in the Lab and in the CM frame. 9~24 In a historic piece of research, James Chadwick in 1932 obtained a value for the mass of the neutron by studying elastic collisions of fast neutrons with nuclei of hydrogen and nitrogen. He found that the maximum recoil velocity of hydrogen nuclei (initially stationary) was 3.3 X l07 m/sec, and that the maximum recoil velocity of nitrogen 14 nuclei was 4.7 X 10“ m/sec with an uncertainty of :l:10%. What does this tell you about (a) The mass of a neutron? (b) The initial velocity of the neutrons used? (Take the uncertainty of the nitrogen measurement into account. Take the mass of an H nucleus as l amu and the mass ofa nitrogen 14 nucleus as 14 amu.) 9~25 A cloud-chamber photograph showed an alpha particle of mass 4 amu with an initial velocity of 1.90 X 107 m/sec colliding with a nucleus in the gas of the chamber. The collision changed the direction of- motion of the alpha particle by 12° and reduced its speed t...
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