Integral calculus
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The document discusses various calculus concepts including: - Indefinite integrals and finding antiderivatives - Definite integrals and properties such as linearity and changing limits of integration - Integrals of trigonometric functions including sine, cosine, tangent, cotangent, secant, and cosecant It provides examples to illustrate each concept discussed.
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1
PHY 4822L (Advanced Laboratory):
Analysis of a bubble chamber picture
Introduction
In this experiment you will study a reaction between “elementary particles” by analyzing their
tracks in a bubble chamber. Such particles are everywhere around us [1,2]. Apart from the standard
matter particles proton, neutron and electron, hundreds of other particles have been found [3,4],
produced in cosmic ray interactions in the atmosphere or by accelerators. Hundreds of charged
particles traverse our bodies per second, and some will damage our DNA, one of the reasons for the
necessity of a sophisticated DNA repair mechanism in the cell.
2
Figure 1: Photograph of the interaction between a high-energy π--meson from the Berkeley
Bevatron accelerator and a proton in a liquid hydrogen bubble chamber, which produces two neutral
short-lived particles Λ0 and K0 which decay into charged particles a bit further.
Figure 2: illustration of the interaction, and identification of bubble trails and variables to be
measured in the photograph in Figures 3 and 4.
The data for this experiment is in the form of a bubble chamber photograph which shows bubble
tracks made by elementary particles as they traverse liquid hydrogen. In the experiment under
study, a beam of low-energy negative pions (π- beam) hits a hydrogen target in a bubble chamber.
A bubble chamber [5] is essentially a container with a liquid kept just below its boiling point (T=20
K for hydrogen). A piston allows expanding the inside volume, thus lowering the pressure inside
the bubblechamber. When the beam particles enter the detector a piston slightly decompresses the
liquid so it becomes "super-critical'' and starts boiling, and bubbles form, first at the ionization
trails left by the charged particles traversing the liquid.
The reaction shown in Figure 1 shows the production of a pair of neutral particles (that do not leave
a ionized trail in their wake), which after a short while decay into pairs of charged particles:
π - + p → Λo + Ko,
3
where the neutral particles Λo and Ko decay as follows:
Λo → p + π-, Ko → π+ + π-.
In this experiment, we assume the masses of the proton (mp = 938.3 MeV/c2) and the pions (mπ+ =
mπ- = 139.4 MeV/c2) to be known precisely, and we will determine the masses of the Λ0 and the K0,
also in these mass energy units.
Momentum measurement
In order to “reconstruct” the interaction completely, one uses the conservation laws of (relativistic)
momentum and energy, plus the knowledge of the initial pion beam parameters (mass and
momentum). In order to measure momenta of the produced charged particles, the bubble chamber is
located inside a magnet that bends the charged particles in helical paths. The 1.5 T magnetic field is
directed up out of the photograph. The momentum p of each particle is directly proportional to the
radius of curvature R, which in turn can be calculated fro.
1PHY 4822L (Advanced Laboratory) Analysis of a bu.docx
The concepts that best describe the situation are unemployment and economic hardship.
Unemployment refers to the father losing his job at the restaurant due to workforce reductions during the pandemic. This has led to economic hardship for the family as they now have limited resources with only one income.
To survive their present condition, the family will need to cut down expenses, practice budgeting, and find alternative sources of income. The father should look for new job opportunities while other family members can consider side jobs or freelance work from home. They may also need to dip into their savings or sell assets if required as a temporary measure until their financial situation improves. Applying for government assistance is another option the family could explore. With prudent financial management and additional
Advanced Algebra 2.1&2.2
1. The formula for inverse variation is y = k/x^n, where k is a non-zero constant and n is greater than 0.
2. For inverse variation, when one variable increases the other variable decreases, and vice versa.
Similar to Integral calculus (20)
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1PHY 4822L (Advanced Laboratory) Analysis of a bu.docx
1PHY 4822L (Advanced Laboratory) Analysis of a bu.docx
1PHY 4822L (Advanced Laboratory) Analysis of a bu.docx
1PHY 4822L (Advanced Laboratory) Analysis of a bu.docx
Integral calculus
- 2. - the process of finding the areas bounded by curves, volumes of various solids, center of gravity and other geometric and physical applications.
- 6. 1. example:
- 7. 2. example:
- 8. 3. example:
- 10. example:
- 12. 1. example:
- 13. 2. example:
- 14. 3. example:
- 15. 4. example:
- 17. example:
- 19. 1. example:
- 20. 2. example:
- 21. 3. example:
- 23. 1. 2. 3. 4.
- 24. 5. 6.
- 25. 7. 8. 9. 10.
- 27. Sine and Cosine Functions
- 28. 1. n≠ -1 n= -1 example:
- 29. b.
- 30. 2. n≠ -1 n= -1 example: a.
- 31. b.
- 32. 3. m is odd example:
- 33. 4. m is odd example:
- 34. example: b.
- 35. 5. m and n are even example: a.
- 36. 6. a. example:
- 37. b. example:
- 38. c. example:
- 39. Tangent and Secant Functions
- 40. 1. m and n are odd example:
- 41. 2. n is even example:
- 42. 3. m is even example:
- 43. Cotangent and Cosecant Functions
- 44. 1. m and n are odd example:
- 45. 2. n is even >2 example:
- 53. , ,
- 64. 1. The integral of a constant multiple of a function is that same constant times the integral of the functions. example:
- 65. 2. The integral of the sum of two functions is the sum of their integrals. example:
- 66. 3. The sign of an integral changes when its limits of integration interchanged. example: