Answer:
Chapter 1
1. Show that the Lorentz transformation is such that the velocity of a light ray
travelling in the x direction is the same for the observer in the frame S and for
the observer in the frame S
.
Solution: Consider a light ray travelling in the x direction. If the light ray
connects two space–time points {t1, x1} and {t2, x2}, we have
c = x2 − x1
t2 − t1
The speed of light observed in the frame S will be
c = x
2 − x
1
t
2 − t
1
= c
γ ((x2 − x1) − βc(t2 − t1))
γ (c(t2 − t1) − β(x2 − x1))
= c
x2 − x1
t2 − t2
− βc
c − β x2 − x1
t2 − t2
= c
2. What is the mean path before decay for a charged pion with a kinetic energy of
1 GeV?
Solution: The pion has a lifetime 2.6 × 10–8 s and a mass of 139.6 MeV. If the
energy is 1 GeV, the velocity of the pion is 99% of the velocity of light (Eq. 1.4).
The mean path before decay is
= 0.99 c γ τ
= 0.99 c
1000 + 139.6
139.6
2.6 10−8 = 63 m
S. Tavernier, Experimental Techniques in Nuclear and Particle Physics, 271
DOI 10.1007/978-3-642-00829-0, C Springer-Verlag Berlin Heidelberg 2010
272 Solutions to Exercises
3. Show that the relativistic expression for the kinetic energy of a particle (Eq. 1.2)
reduces to the non-relativistic expression if the velocity of the particle is small
compared to the velocity of light.
Solution:
E = Ekinetic + m0c2 = m0c2
1 − (v/c)2
≈ m0c2
(1 − 1/2(v/c)2) ≈ m0c2(1 + 1/2(v/c)
2)
= m0c2 +
1
2
m0v2
4. For a Poisson distribution with average value 16, calculate the probability to
observe 12, 16 and 20 as measured value. Calculate the probability density function for a Gaussian distribution with average value 16 and dispersion 4, for the
values x = 12, 16 and 20. Compare the results.
Solution: For a Poisson distribution P(12) = 0.0829, P(16) = 0.1024, P(20) =
0.0418
For a Gaussian distribution, f(12) = 0.0605, f(16) = 0.0997, f(20) = 0.0605
5. Consider a very short-lived particle of mass M decaying into two long-lived particles 1 and 2. Assume you can measure accurately the energies and momenta of
the two long-lived particles. How will you calculate the mass of the short-lived
particle from the known energies and momenta of the two long-lived objects?
Solution: The mass of the short-lived particle, its energy and its momentum are
related by Eq. (1.1). The energy and momentum of the particle are equal to the
sums of the energy and sums of the momenta of the decay products, therefore
M2c4 = (E1 + E2)
2 − c2(P1 + P2)
2
6. Calculate the order of magnitude of the energy levels in atoms and in nuclei
using the ‘particle in a box’ approximation, Eq. (1.9). Use for the dimension of
the atom 10–10 m and for the dimension of the nucleus 10−15 m.
Solution: Atomic energy levels: ≈40 eV; nuclear energy levels: ≈400 MeV.
7 . Show that in a β− or a β+ decay only a very small fraction of the energy derived
from the mass difference goes to the kinetic energy of the final-state nucleon.
The electron is relativistic; therefore this requires a relativistic calculation! Hint:
the 3-body problem can be reduced to a 2-body problem by considering the
electron–neutrino system as one object with a mass of a few MeV.
Solution. Consider the 2-body decay of some heavy object with mass M into
two objects with masses m1 and m2. The kinetic energy of each of the final-state
particles in the overall centre of mass system is found as follows.
Solutions to Exercises 273
Consider two particles with energy and momentum four vectors p1 and p2.
The symbol pi stands for the four-vector {Ei,cpi}. The energy E appearing in this
expression is the total energy E, i.e. the rest energy mc2 plus the kinetic energy.
The four-vector product (p1.p2) is defined as
(p1.p2) =
(
E1E2 − c2 p1 p2
)
A four-vector product is a Lorentz invariant; this quantity can be evaluated in
any reference frame, and the result is the same. Consider now the quantity
(p1.p2)
m1c2
This is a Lorentz invariant. Evaluating this expression in the rest frame of
particle 1 makes clear that this is the energy of particle 2 seen in the rest frame
of particle 1. This remains true also if one of the particles is in fact a system
of particles, for example the system of the two particles 1 and 2. The energy of
particle 2, seen in the overall centre of mass frame of the particles 1 and 2 is
therefore
E∗
2 = (p1 + p2).p2
(p1 + p2)2
We have the following relations:
(p1 + p2)
2 = M2c4
(p1.p2) = 1
2
(
(p1 + p2)
2 − (p1)
2 − (p2)
2
)
= M2c4 − m2
1c4 − m2
2c4
And therefore finally
E∗
2 = M2c4 + m2
2c4 − m2
1c4
2Mc2
Let us now apply the above relation to the decay
N∗ → N + e− + ¯νe + Q
The symbol Q represents the energy liberated in the reaction. Let us denote
by M∗ the mass of the parent nucleus, by M the mass of the final-state nucleus
and by m the mass of the electron–neutrino system. The kinetic energy of the
nucleus in the final state is given by
274 Solutions to Exercises
Ekin = M∗2c4 + M2c4 − m2c4
2M∗c2 − Mc2
= M∗2c4 + M2c4 − m2c4 − 2M∗c2Mc2
2M∗c2
= (M∗ − M)
2 c4 − m2c4
2M∗c2
=
!
mc2 + Q
The energy of the cosmic ray photon is zero. his means that the photon had insufficient energy to create new particles, and instead, it simply scattered off the massive object.
What is Einstein's energy equation?Einstein's energy equation, also known as the mass-energy equivalence, relates the energy E of an object to its mass m and the speed of light c. The equation is:
E = mc^2
where:
E is the energy of the object in joules (J)
m is the mass of the object in kilograms (kg)
c is the speed of light in meters per second (m/s)
This equation means that mass and energy are interchangeable, and that a small amount of mass can be converted into a large amount of energy. The equation is an important consequence of Einstein's theory of special relativity, and it has been confirmed by numerous experiments, including nuclear reactions and particle accelerators.
Here in the Question,
We can use the conservation of momentum and energy to solve this problem. Since the two fragments have equal mass and are moving in opposite directions at the same speed, we know they have equal and opposite momenta. Therefore, the initial momentum of the photon must also be equal and opposite to the total momentum of the fragments.
Let's call the initial momentum of the photon p and the mass of each fragment m. The total momentum of the fragments is:
p' = 2mv
where v is the speed of each fragment, which we know is 0.6c. Therefore, we can write:
p' = 2m(0.6c) = 1.2mc
By conservation of momentum, we have:
p = -p'
where the negative sign indicates that the photon is moving in the opposite direction to the fragments. Therefore:
p = -1.2mc
Now we can use conservation of energy to relate the photon's energy E to its momentum p:
E^2 = p^2c^2 + m^2c^4
Substituting the expression we found for p, we get:
E^2 = (1.2mc)^2c^2 + m^2c^4
E^2 = 1.44m^2c^4 + m^2c^4
E^2 = 1.45m^2c^4
Solving for E, we get:
E = mc^2 * sqrt(1.45)
Plugging in the values for m and c, we get:
E = (0 * 9.0 × 10^16 kg) * sqrt(1.45) = 0
Therefore, The photon from a cosmic ray has no energy. This indicates that the photon was merely scattered off the large object since it lacked the energy to produce new particles.
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Radioactive isotopes can be used to find the age of rocks, fossils, or other artifacts. Carbon-14 has a half-life of 5,730 years. Suppose a sample of charcoal from a primitive fire pit contains one eighth of its original amount of carbon-14. How old is the sample?
Answer:
1/8 = (1/2)^3
This implies the sample has decayed for 3 half lives
3 * 5730 yrs = 17,200 years
A kitten sits in a lightweight basket near the edge
Answer:
So where's the problem now?
A kitten is just sitting
Mrs. Martin is holding a ball from a height of 3.2 m that has the potential energy of 23 J, calculate the mass of the ball.
Answer:
0.73kg
Explanation:
Potential gravitational energy is equal to [tex]Pe=mgh[/tex], where Pe is potential gravitational energy, m is mass in kilograms, g is gravity in m/s^2 and h is height in meters.
Substitute:
[tex]Pe=mgh\\23=m*9.8*3.2\\23=31.36m\\0.73=m[/tex]
A ball is thrown directly upwards at 25 ms and the same time another ball is released from height of loom. At what height and time will the balls meets?
Answer:
Explanation:
When the balls meets, the time should be the same, let say t.
For the first ball (released from a loom):
[tex]\bar{h}=\frac{1}{2}gt^{2}[/tex] (Free fall motion) ([tex]\bar{h}[/tex] = the distance travelled by the first ball, measured from the loom)
So, we can find that [tex]\bar{h}=\frac{1}{2}(10t^{2})=5t^{2}[/tex]
For the second ball (which is thrown directly upwards):
[tex]y=v_{o}t-\frac{1}{2}gt^{2} = 25t-5t^{2}[/tex]
Because the height of the loom is h :
[tex]h=\bar{h}+y[/tex]
[tex]h=25t[/tex]
So, the time when two balls meets is : [tex]t=\frac{h}{5}[/tex] (In this question, the height of the loom h is not declared). When the height of the loom is known, you can calculate the time and the value of [tex]\bar{h}[/tex] or y
15. A pattern of weather and
climate over three months is called a
Answer:
Explanation:
weather pattern
A pattern of weather over three months is called a climate.
What is climate?Climate is the long-term pattern of weather in a particular area. Weather can change from hour-to-hour, day-to-day, month-to-month or even year-to-year. A region's weather patterns, usually tracked for at least 30 years, are considered its climate.
Weather refers to short-term changes in the atmosphere, climate describes what the weather is like over a long period of time in a specific area.
A pattern of weather over three months is called a climate.
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In the fishbowl, the glass, water, rocks, and plastic plants are in thermal equilibrium. This situation means the temperature of the rocks is...
Which nucleus completes the following equation?
Answer:
I think that CL IS THE ONLY BEST ANSWER
Explanation:
PLEASE MARK ME BRAINLIEST IF MY ANSWER IS CORRECT PLEASE
What is the cycle of a breaststroke?
Answer:
Breaststroke can be divided up into 4 distinct phases, which consists of the out sweep, in sweep, breathing phase, and lunge, extension, or recovery phase.
Explanation:
pa follow po
what are the two most important traits between a whale and a shark?
Answer:
Both are aquatic animals and are hunters
Explanation:
List three methods of energy transport in nature and explain how the energy is being transferred in each of those methods. Which two are means by which energy is transported inside the Sun?
This question is for my college astronomy class!
First normal transportation is the light energy of sun is transmitted to our earth .
Next for transportation inside sun they are
Fusion and fissionFusion means when two hydrogen atoms reacts to form helium nucleus .
Fission means when one nucleus is broken into two or more nucleus
Explanation:
#Make_the smartest_answer
#sorry_if_wrong
#I hope this helps
#follow_and_like_me
#No copas
#Thank_you_for_watching
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ॐhน໓ค • ๓นhค๓๓ค໓ঔৣ
ॐŞiŞ¢ค • วนคriคঔৣ
You perform an experiment to test how far a cannon will shoot a performer at
different angles. The data are shown below.
Angle
Distance (feet)
110
32°
36°
120
40°
128
440
133
48°
127
52
119
56°
109
60°
100
64°
91
68°
77
72°
63
At which angle will the performer travel the longest distance?
Answer:
44°
Explanation:
Look for the longest distance under the 'Distance ' column....look to the left to find the corresponding angle
Table clearly shows that the longest distance (133 ft) occurs at angle of 44°
What is the momentum of a 35–kilogram cart moving at a speed of 1.2 meters/second
Answer:
42
Explanation:
P = M × V
so
P = 35 × 1.2
P = 42
Answer:
42kg•m/s
Explanation:
p=m*v
An unknown fluid flows at a speed of 31 m/s. Suppose the fluid has a mass of 47 kg runs at this speed. What is the fluid’s kinetic energy?
Answer:
22583.5J
Explanation:
KE=1/2 mv^2
=1/2*47Kg*(31m/s^2)
=23.5Kg * 961m/s^2
=22583.5J
A parachute falling to the ground.
Answer:
a parachute falling to the ground is uniform
An aquarium heater transfer 1200kJ of the heat to 75000g of water. What is the increase in the waters temperature?
Answer:
Energy = mcΔt
where m the mass , c specific heat capacity
Δt is the change in temperature
1200x10^3=75x4184xΔt
Δt=3.82°
Explanation:
A rock is suspended from the lower end of a light string. When the rock is totally immersed in water, the tension in the string is 1.20 N. When the rock is totally immersed in ethanol, the tension in the string is 1.60 N. Calculate the weight of the rock.
For a rock is suspended from the lower end of a light string, the weight of the rock is mathematically given as
W = 3.10 N
What is the weight of the rock.?Generally, the equation for the W = Fh1+ is mathematically given as
W = Fh+T1
Where
PiVg+T1 = P2V9+T
(1000) V(9.8) + 1.2 = (790) V(9.8) + 1.6
V = 1.944* 10^-4 m3
In conclusion
W = Fh+T1
W = (1000) (1.944 * 10-4) (9.8) + 1.2
W = 3.10 N
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list down the general of the office of guardian and tutor
Answer:
legal guardians, the four types: personal guardians, financial guardians, conservators, and pet “guardians”
In addition to their remarkable top speeds of almost 60 mph, cheetahs have impressive cornering abilities. In one study, the maximum centripetal acceleration of a cheetah was measured to be 18 m/s2. What minimum value of the coefficient of static friction between the ground and the cheetah's feet is necessary to provide this acceleration?
coefficient of static friction is 1.7329
Explanation:
given data
velocity = 60 mph
acceleration = 17 m/s²
to find out
coefficient of static friction
solution
we will apply here centripetal force equation
that is
m×v²/r = µ × m × g ..................1
here v²/r is centripetal acceleration and m is mass and µ is coefficient static friction so
µ = a / g
µ = 17 / 9.81
µ = 1.7329
so coefficient of static friction is 1.7329
I need help with this please
Answer:
i think its 100
Explanation:
For the statements based on whether they are described as outcomes of results from thermal energy being added or removed particles move faster particles move slower temperature increases temperature decreases kinetic energy increases kinetic energy decreases thermal energy added thermal energy removed
If you have a mass of 0.9kg and a velocity 12 m/s what is your momentum?
A turntable is switched from 11.3 rad/s to 1.9 rad/s, and the platter goes through an angle of 17.95 radian in reaching the new angular speed. What is the angular acceleration of the platter?
Hi there!
We can use the angular equivalent of the following kinematic equation:
[tex]\omega_f^2 = \omega_i^2 + 2\alpha \theta[/tex]
ωf = final angular velocity (1.9 rad/s)
ωi = initial angular velocity (11.3 rad/s)
α = angular acceleration (? rad/s²)
θ = angular displacement (17.95 rad)
We can rearrange the equation to solve for angular acceleration.
[tex]\omega_f^2 - \omega_i^2 = 2\alpha \theta\\\\\alpha = \frac{\omega_f^2 - \omega_i^2}{2\theta}[/tex]
Plug in the given values and solve.
[tex]\alpha = \frac{1.9^2 - 11.3^2}{2(17.95)} = \boxed{3.456 \frac{rad}{s}}[/tex]
The current in the wires of a circuit is 60.0 milliAmps. If the
voltage impressed across the ends of the circuit were tripled
(with no change in its resistance), then its new current would
be milliAmps.
Answer:
180 m amp
Explanation:
V = IR
v/r = i mutiply both sides by
3
3v / r = 3 i so the current will triple to 180 mAmp
A garage hoist lifts a truck up 2 m above the ground in 15seconds.Find the power delivered to the truck (given 1000kg as the mass of the truck and gas 9.81m/s
The power delivered to the truck of mass 1000 kg that was lifted by a garage hoist, 2 m high above the ground in 15 seconds is 1308 W.
What is power?
Power can be deifned as the rate at which work is done.
To calculate the power delivered to the truck, we use the formula below.
Formula:
P = mgh/t.............. Equation 1Where:
P = Power delivered to the truckm = Mass of the truckh = Heightg = Acceleration due to gravityt = timeFrom the question,
Given:
m = 1000 kgh = 2 mt = 15 secondsg = 9.81 m/s²Subsitute these values into equation 1
P = (1000×2×9.81)/15P = 19620/15P = 1308 WHence, The power delivered to the truck of mass 1000 kg that was lifted by a garage hoist, 2 m high above the ground in 15 seconds is 1308 W.
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The power in an electrical current is given by the equation
Answer:
P = VI
Explanation:
the power is equal to the current × voltage
Answer:
P = V • I
Explanation:
Power = Voltage • Current
A total charge of Q coulomb is uniformly distributed along a rod 40cm in length.find find the electric field intensity 20cm away from the rod along its perpendicular bisector?
For a total charge of Q coulomb is uniformly distributed along a rod 40cm in length, the electric field intensity 20cm away from the rod is mathematically given as
E1=1.598*10^11v/m
Generally, the equation for the initial electric field intensity is mathematically given as
[tex]dEp=\int{kd/r}cosd\theta[/tex]
Therefore
[tex]e1=kd/r{sin\theta2+sinR1}[/tex]
Hence
[tex]E1=(9*10^9/20*10^{-2})({sin45+sin45})*B/40*10^-2[/tex]
E1=B*9*10^{13})/(10*110)*[tex]\sqrt{2}[/tex]
E1=1.598*10^11v/m
In conclusion, the electric field intensity
E1=1.598*10^11v/m
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Residents of a community currently pay a fee for recreational services in their neighborhood, such as tennis courts, running trails, a pool. To assess whether there is support for raising a fee for the upcoming summer, a random sample of 50 residents is contracted and asked this question: “Would you support a $10 increase in the recreational fee to support programs for the health and well-being of the children in the community?” Of the 50 residents contacted, 42 approved of the fee increase. Which statement describes a likely source of bias present in survey results?
A likely source of bias in the survey is undercoverage bias because some parts of the community may not be represented in the survey results.
Bias present in the surveyThe bias present in the survey may lead to error in the result obtained from the random sampling survey.
Source of bias in the surveyOne of the likely source of bias in the survey is undercoverage.
It is not possible to ascertain the accurate opinion of the entire community from the 50 residents chosen for the survey.
Thus, a likely source of bias present in survey is undercoverage bias because some parts of the community may not be represented in the survey results.
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Answer:
B. Question wording bias may be present because the question leads those responding toward a certain response.
Explanation:
A potter's wheel is spinning with an initial angular velocity of 11 rad/s . It rotates through an angle of 80.0 rad in the process of coming to rest.
What was the angular acceleration of the wheel?
How long does it take for it to come to rest?
Hi ! I will help you to discuss about Proportionally Changes in Circular Motion. The analogy of proportionally changes in circular motion is same as the analogy of proportionally changes in direct motion. Here you will hear again the terms acceleration and change in speed, only expressed in the form of a certain angle coverage. Before that, in circular motion, it is necessary to know the following conditions:
1 rotation = 2π rad1 rps = 2π rad/s1 rpm = [tex] \sf{\frac{1}{60} \: rps}[/tex] = [tex] \sf{\frac{1}{30}\pi \: rad/s}[/tex] Formula UsedThe following equations apply to proportionally changes circular motion:
Relationship between Angular Acceleration and Change of Angular Velocity[tex] \boxed{\sf{\bold{\omega_t = \omega_0 + \alpha \times t}}}[/tex]
With the following conditions:
[tex]\sf{\omega_t}[/tex] = final angular velocity (rad/s)[tex]\sf{\omega_0}[/tex] = initial angular velocity (rad/s)[tex]\sf{\alpha}[/tex] = angular acceleration (rad/s²)t = interval of the time (s)Relationship between Angular Acceleration and Change of [tex]\sf{\theta}[/tex] (Angle of Rotation)[tex] \boxed{\sf{\bold{\theta = \omega_0 \times t + \frac{1}{2} \times \alpha \times t^2}}}[/tex]
Or
[tex] \boxed{\sf{\bold{(\omega_t)^2= (\omega_0)^2 + 2 \times \alpha \times \theta}}}[/tex]
With the following condition :
[tex]\sf{\theta}[/tex] = change of the sudut (rad)[tex]\sf{\alpha}[/tex] = angular acceleration (rad/s²)t = interval of the time (s)[tex]\sf{\omega_t}[/tex] = final angular velocity (rad/s)[tex]\sf{\omega_0}[/tex] = initial angular velocity (rad/s) Problem SolvingWe know that :
[tex]\sf{\omega_t}[/tex] = final angular velocity = 0 rad/s >> see in the sentence "in the process of coming to rest."[tex]\sf{\omega_0}[/tex] = initial angular velocity = 11 rad/s[tex]\sf{\theta}[/tex] = change of the sudut = 80.0 radWhat was asked :
[tex]\sf{\alpha}[/tex] = angular acceleration = ... rad/s²t = interval of the time = ... sStep by step :
[tex]\sf{\alpha}[/tex] = ... rad/s²[tex] \sf{(\omega_t)^2= (\omega_0)^2 + 2 \times \alpha \times \theta} [/tex]
[tex] \sf{0^2= (11)^2 + 2 \times \alpha \times 80} [/tex]
[tex] \sf{0 = 121 + 160 \alpha} [/tex]
[tex] \sf{-160 \alpha = 121} [/tex]
[tex] \sf{\alpha = \frac{121}{-160}}[/tex]
[tex] \sf{\alpha = -0.75625 \: rad/s^2 \approx \boxed{-0.76 \: rad/s^2}}[/tex]
t = ... s[tex] \sf{\alpha = \frac{\omega_0 - \omega_t}{t}}[/tex]
[tex] \sf{-0.76 = \frac{0 - 11}{t}}[/tex]
[tex] \sf{-0.76t = -11}[/tex]
[tex] \sf{t = \frac{- 11}{-0.76}}[/tex]
[tex] \boxed{\sf{t \approx 14.474 \: s}}[/tex]
ConclusionSo :
The angular acceleration of the wheel approximately -0.76 rad/s² or proportionally as deceleration approximately 0.76 rad/s.It need approximately 14.474 s to come to rest.if a copper coin with a volume of exactly 50 cubic cm is dropped into the volume of water 4cubic cm,how much water will be overflown (water is half way of the can)
Answer:
50 cubic cm
Explanation:
As the water displaces 50 cubic centimeters the amount of water that must leave the container for it to be able to be dropped into it is also 50 cubic as the water which was in its place must have gone somewhere.
Assume that the body's muscle mechanism can be approximated by a spring with a uniform continuous mass distribution that follows Hooke's law. Concerning this,
A) find the effective mass of the spring with mass m.
Then, estimate the potential energy which can be mechanically stored in B) the muscles of each arm, and
C) the muscles of each leg,
and estimate the spring constant of
D) each arm muscles, and
E) each leg muscle.
F) Now, could estimate the speed of a runner by using these results?
Based on Hooke's law, the spring constant of the the body's muscle mechanism is the ratio of force to extension, the effective mass is m/3 and the potential energy that can be stored is ke^2 / 2.
What is the spring constant?The spring constant or stiffness constant of an elastic spring is constant which describes the extent a bit forceapplied to an elastic spring will extend it.
Spring constant, K = force/extensionAssuming, a body's muscle mechanism is a spring obeying Hooke's law, the effective mass of the spring with mass m is 1/3 of the mass of the spring = m/3
The potential energy that can be stored = ke^2 / 2
where K is spring constant and e is the extension produced.
Therefore, the spring constant of the the body's muscle mechanism is the ratio of force to extension, the effective mass is m/3 and the potential energy that can be stored is ke^2 / 2.
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