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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 cloudchamber 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 multiplestage 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 twostage
rocket (see the ﬁgure). The total mass—~both rockets fully fueled, plus . the payload—is Nm. The mass of the secondstage rocket plus the payload, after ﬁrststage 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 ﬁrststage 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 secondstage 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 singlestage
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 twostage rocket but is im—
possible, however large the value of N, with a singlestage 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 crosssectional 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 onedimensional
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 8kg 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 8kg 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 8kg 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 centerof
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). 923 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 cloudchamber 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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