Fisika Rumus Untuk Olimpiade Fisika Internasional

Formulas for IPhO I Mathematics 1. Taylor’s series:

2. If in the problem’s text is an extraordinary co- the points, are equal; if one connects two points incidence (e.g. two things equal), then look for with a line and projects the velocities of these 7. Inscribed angles with identical endpoints are the solution’ s key there. points on the drawn line, then the projections of equal. They equal half of the corresponding cent3. Read carefully the recommendations in prob- the velocities equal; b) the instantaneous center ral angle. lem’ s text. Pay attention to problem’s formulation of rotation can be found via two velocity vectors Conclusions: hypotenuse of a right triangle is the diameter of circumventer; if the sum of quad- — sometimes insignificant details may carry vi- (either from the intersection point of perpendicrangle’s opposite angles is a straight angle, then tal information. If you have solved for some time ulars or from similar triangles). a/ sin α = b/ sin β = 2R

X F (x) = F (x0 ) + F (n) (x − x0 )n /n! Special case — linear approximation: F (x) ≈ F (x0 ) + F ′ (x − x0 ) it’s apexes lie on a circumference. Applications are simpler, if x ≪ 1: 2 x 8. Taking derivatives: sin x ≈ x, cos x ≈ 1 − x /2, e ≈ 1 + x (f g)′ = f g ′ + f ′ g, f [g(x)]′ = f ′ [g(x)]g ′ ln(1 + x) ≈ x, (1 + x)n ≈ 1 + nx 2. Perturbation method: next and more precise (sin x)′ = cos x, (cosx)′ = − sin x solution is based on the previous one. (ex )′ = ex , (ln x)′ = 1/x, (xn )′ = nxn−1 3. Solutions of the linear differential equation (arctan x)′ = 1/(1 + x2p ) ′′ ′ ′ ′ with constant coefficients ay + by + cy = 0 (arcsin x) = −(arccos x) = 1/ 1 − x2 can be represented as: 9. Integration: the formulas are the same as for y = A exp(λ1 x) + B exp(λ2 x), derivation, but in reverse order (inverse operawhere λ1,2 is the solution of the so called charac- tion!), e.g. Z teristic equation aλ2 + bλ + c = 0 if λ1 6= λ2 . xn dx = xn+1 /(n + 1). If the solution of the characteristic equation is complex, while a, b and c are real numbers, then Special case of substitution method: Z λ1,2 = γ ± iω and f (ax + b)dx = F (ax + b)/a. y = Ceγx sin(ωx + ϕ0 ).

4. Complex numbers z = a + bi = |z|eiϕ , z¯ = a − ib = |z|e−iϕ b |z|2 = z z¯ = a2 + b2 , ϕ = arg z = arcsin |z| Rez = (z + z¯)/2, Imz = (z − z¯)/2 |z1 z2 | = |z1 ||z2 |, arg z1 z2 = arg z1 + arg z2 eiϕ = cos ϕ + i sin ϕ iϕ 2 cos ϕ = e + e−iϕ , 2i sin ϕ = eiϕ − e−iϕ 5. Vector and scalar products of vectors are distributive: a(b + c) = ac + bc. Cross product is anticommutative: ~a × ~b = −~b × ~a. ~a · ~b = ab cos ϕ, |~a × ~b| = ab sin ϕ. Definition of the mixed product: (~a, ~b, ~c) ≡ (~a · [~b × ~c]) = ([~a × ~b] · ~c). In case of double vector product: ~a × [~b × ~c] = ~b(~a · ~c) − ~c(~a · ~b).

6. Cosine and sine laws:

c2 = a2 + b2 − 2ab cos ϕ

unsuccessfully, then read the text again — perhaps you misunderstood the problem.

5. Non-inertial reference system: ~ + ~aCor ~v2 = ~v0 + ~v1 , ~a2 = ~a0 + ~a1 + ω 2 R

4. Make long and time-consuming transforma- 6. Ballistic problem: tions then, when everything else is done. BUT always write down all initial expressions, what need to be transformed.

5. If the problem seems to be hopelessly diffi-

y ≤ v02 /(2g) − gx2 /2v02 .

IV Dynamics

rd cult, then it usually has an extremely simple solu- 1. Newton’s 3 law for translational and rotation (and a simple answer). This is valid only for tional motion: ~ = I~ε (M ~ = ~r × F~ ). F~ = m~a, M Olympiad problems, because they definitely have ~ and ~ε are basically a solution. In case of 2D geometry M 6. In experiments a) sketch schema of the ex- scalars and M = F l, where l is arm of a force. periment; b) think, how to increase the precision 2. Newton’s 3rd law: if system’s state is deof the results. scribed by one parameter ξ, its derivative over time ξ,˙ potential energy Π(ξ) and kinetic energy K = µξ˙2 /2, then µξ¨ = −dΠ(ξ)/dξ. ConcluIII Kinematics 10. Numerical methods sion: force is potential energy derivative over co— derivatives, integrals: Newton’s formula for finding function’s 1. Translational motion Z Z ordinate. d~x f (x) = 0 zero crossing points: ~v = , ~x = ~v dt (x = vx dt jne.) 3. If system consists of mass points mi : dt X X X Z xn+1 = xn − f (xn )/f ′ (xn ). 2 ~ r = mi~ri / mj , P~ = mi~vi d~v d ~v c ~a = = 2 , ~v = ~adt X X Trapezoidal rule: dt Z dt ~ = Z Z L mi~ri × ~vi , K = mi vi2 /2 Z b vx Z −1 −1 a−b X t = vx dx = ax dvx , x = dvx f (x)dx ≈ [f (x0 ) + 2f (x1 ) + . . . ax Iz = mi (x2i + yi2 ) = (x2 + y 2 )dm. 2n a If a = Const., then previous integrals can be +2f (xn−1 ) + f (xn )] 4. In a system, which moves relative to mass found easily, e.g. 2 2 2 11. Derivatives and integrals of vectors: either center with velocity ~vc (index c marks system rex = v0 t + at /2 = (v − v0 )/2a. by components or (in case of derivative) geometlated to mass center): 2. Rotary motion — analogy to translational ~ =L ~ c + MΣ R ~ c × ~vc , K = Kc + MΣ vc2 /2 rically — according to triangle rule. L motion; e.g. ω = dϕ/dt, ε = dω/dt;

II General recommendations

~a = ~τ dv/dt + ~nv 2 /R

5. Steiner’s theorem (b — distance of point mass

2 3. Curvilinear motion — same as point 1, but from rotational axis): I = Ic + mb . 1. Check all formulas for veracity: a) exam- vectors must be replaced by linear velocities, ac- 6. Changeover from one reference system to an-

other: P~ = P~c + MΣ~vc ine dimensions; b) test simple special cases (e.g. celerations and path lengths. asymptotes); c) verify the plausibility of solu- 4. Motion of rigid body: a) projections of velo- 7. For a system as a whole: ~ Σ = dL/dt ~ tion’s qualitative properties. cities of two points on the line, which connects F~Σ = dP~ /dt, M

8. Inertial momentum relative to mass cen- if sliding stops, then the velocities of collision ter and z-axis can also be calculated so: Iz0 = planes of both impacting bodies were equal in P 2 2 collision point; d) if sliding does not stop, then i,j mi mj [(xi − xj ) + (yi − yj ) /2MΣ ]. 9. Inertial momentum relative to the origin of the momentum is delivered relative to surface normal under angle arctan µ. coordinates: 2θ = Ix + Iy + Iz .

3. N coupled oscillators have N natural oscillations ωi (all oscillators oscillate with same frequency ωi , but with different amplitudes: xj = xj0 sin(ωi t + ϕj ). General solution is superposition of all natural oscillations and contains opX and φi : 16. Every motion of rigid body can be represen- tional constants 10. Physical pendulum: X i 2 2 2 xj = Xi xj0 sin(ωi t + ϕj + φi ) ted as rotation around the instantaneous center ω (l) = g/(l + q b /l), b = I/m j 2 of rotation (in the sense of velocity field). NB! ω(l) = ω(˜l − l) = g/˜l, ˜l = l + b /l 11. Coefficients of inertial momentum: cylinder Distance of the point from the instantaneous 4. If system can be described by one2parameter 1/2, solid sphere 2/5, thin spherical shell 1/3, center of rotation is not its radius of curvature ξ and it is known, that Π(ξ) = κξ /2 [where of trajectory. κ = Π′′ (0)] and K = µξ˙2 /2, then ω 2 = κ/µ. rod 1/12 (about end 1/3), square 1/6. 17. Tension in a string: in case of a massive 5. If in a fixed point wave’s frequency is ν and 12. Often applicable conservation laws: string hanging, tension’s horizontal component wavelength is λ, then phase velocity is energy (elastic, no friction), vf = νλ = ω/k. momentum (sum of external forces is zero; also is constant and vertical changes according to the string’ s mass underneath. Pressure force only in the direction of one axis), 6. For linear waves (electromagnetic waves, angular momentum (if sum of external torques of a string resting on a smooth surface is de- sound and surface waves with small amplitude) is zero, e.g. there are no external forces or their termined by its radius of curvature and tension: arbitrary wavelet is superpositions of single siarm of force is zero; also applicable with respect N = T /R. Analogy: surface tension pressure nusoidal waves. Standing wave is the sum of two p = 2σ/R; to derive, investigate pressure force to one point). identical waves traveling in opposite directions. along diameter. 13. Non-inertial reference systems: in them acts 7. Speed of sound in gas p p p additional inertial force m~a, centrifugal force 18. Adiabatic invariant: if parameters of a peri= (∂p/∂ρ) = γp/ρ = v ¯ γ/2. c s adiab ~ and Coriolis force (is zero, if body is at odically moving system change very little during mω 2 R 8. Speed of sound in elastic material cs = standstill; work is zero, because it is perpendicu- one period, then the area of the shape on phase p E/ρ. plane (in p-x coordinate system) is very precisely lar to velocity). 9. Doppler’s effect: if cs ≫ vk , then 14. Tilted coordinate system: in case of an in- constant (though not absolutely constant). 19. For investigating stability use a) principle ∆ν = ν0 vk /cs . clined plane it is often practical to choose axes of minimum potential energy or b) principle of along and perpendicular to the plane; gravita10. Huygens’ principle: wavefront can be contional acceleration has then both x- and y- com- small virtual displacement. structed step by step, placing an imaginary wave ponents. Axes may not be perpendicular, but 20. Virial theorem for finite movement: source in every point of previous wave front. Resthen are finding the components of a vector in a) F ∝ |~r|, then hKi = hΠi; ults are curves separated by distance ∆x = the direction of axes and projecting the vector to b) F ∝ |~r|−2 , then 2 hKi = − hΠi. cs ∆t, where ∆t is time step and cs is the veloaxes very different procedures. city in given point. Waves travel perpendicular to wavefront. 15. Collision of bodies: maintained are a) total V Oscillations and waves momentum, b) total angular momentum, c) angular momentum of each body relative to point 1. Dissipated oscillator: VI Geometrical optics. Photometry. x ¨ + 2γ x˙ + ω02 = 0 (γ < ω0 ). of collision immediately before and after the col1. Fermat’s principle: waves path from point A lision (for the last body, this equation could be Solution of this equation is (look I.2.): q to point B is such, that the wave travels the least deduced from previous equations [a)–c)], d) total x = x0 e−γt sin( ω02 − γ 2 t − ϕ0 ). time. energy (for elastic collision); kinetic energy may be maintained only along one axis (elastic col- 2. Coupled oscillator’s general equation of mo- 2. Snell’s law: P lision with friction). Additional equations: e) tion: x ¨i = i aij xj . sin α1 / sin α2 = n2 /n1 = v1 /v2 .

3. If refraction index changes continuously, then we imaginarily divide the media into layers of constant n and apply Snell’s law. Light ray can travel along a layer of constant n, if the requirement of total internal reflection is fulfilled, n′ = n/r (where r is curvature radius). 4. If refraction index depends only on zcoordinate, then kx , ky = Const., |~k|n = Const.

5. The thin lens equation (pay attention to signs): 1/a + 1/b = 1/f = D. 6. Newton’s equation (binds distances of image and original from focal plane): x1 x2 = f 2 . 7. Parallax method for determination of the image position: find auxiliary body’s position where it is at standstill, if viewpoint shifts relative to image. 8. Geometrical constructions for finding path of light ray through lenses: a) ray passing through center of lens does not refract; b) ray parallel to optical axis (or its elongation) passes through focus; c) parallel rays converge in focal plane; d) Image of a plane through a lens is a plane, image of a line is a line and image of a point is a point. Line and elongations of its image converge in lens’ plane. 9. Luminous flux Φ [unit is lumen (lm)] characterizes the intensity of light, which is perceived by the eye, while light passes through some (imaginary) surface. Luminous intensity [unit is candela (cd)] is luminous flux radiated by light source into solid angle: I = Φ/Ω. Illuminance [unit is lux (lx)] is luminous flux incident upon a surface per unit area: E = Φ/S. 10. Gauss theorem for luminous flux: if imaginary surface surrounds point sources of light, that radiate equally in every direction, then Φ =

P 4π Ei ; illuminance irradiated by point source of light E = I/r2 .

7. Reflection from optically denser dielectric media: phase shift π.

11. If a grease stain on a paper is as bright as 8. Fabri-Perot interferometer: two parallel the surrounding paper, then the paper is equally semitransparent mirrors with reflection index r, 1 − r ≪ 1. Width of transmission spectra is illuminated from both sides. λ(1 − r). Can be derived by a) adding reflections, reflections of reflection, etc. (geometric progresVII Wave optics sion) or b) finding amplitudes of waves traveling 1. Diffraction — general method (is based on in opposite directions from limit condition. Huygens’ principle): if obstacles divide wavefront 9. Coherent electromagnetic waves: electric into one or more pieces of plane wave, then the fields are added; vector diagram can be used, pieces of wavefront can be filled by imaginary angle between vectors is phase shift; NB! refraclight sources and their interference can be in- tion index n = n(ω). vestigated. I ∝ nE 2 . 2. Two slit interference (width of the slit is 10. Malus’ law: for linearly polarized light I = a ≪ d): angles of maximum are sin ϕmax k = I0 cos2 ϕ, where ϕ is angle between planes of pokλ/d, k ∈ Z. larization. 3. Single slit with width d: angles of minimum 11. Brewster’s angle: reflected and refracted are sin ϕmin k = kλ/d, k ∈ Z, k 6= 0. Central rays are perpendicular; reflected ray is totally pomaximum is twice as wide. To derive divide the larized; incident angle slit imaginarily into halves, quarters, eighths etc.; ϕB = n. look point 1. 12. Newton’s biprism: if diffraction picture is to be considered, then lenses and prisms can be of main maximums is the same as in point 2. Diffraction grating is analyzed the same way neglected and only images investigated. as single slit interference. Spectral resolution (waves with wavelengths λ and λ + ∆λ can be VIII Circuits distinguished; N is the number of slits and k — 1. U = IR, P = U I. diffraction order of main maximum): 2. Kirchoff ’s laws: ∆λ = λ/kN. X X I = 0, U =0 5. Resolution of an ideal telescope (lens): points contuor node are distinguishable, if angle between them is 3. Methods of solution: a) method of potentials; 4. Diffraction grating with step d: location

ϕ = 1,22λ/d.

ϕ = arg Z, Ueff = |Z|Ieff X P = |U ||I| cos(arg Z) = Ii2 Ri .

6. Characteristic times:

√ τRC = RC, τLR = L/R, ωLC = 1/ LC.

Approach to stationary current distribution happens exponentially, ∝ e−t/τ .

7. Energy conservation for electric circuits: ∆W + Q = U q,

ZR = R, ZC = 1/iωC, ZL = iωL;

3. Circulation theorem I

~ ~l = 0 (= Φ), ˙ Ed

I ~ I Bdl = I, ~gd~l = 0. µµ0

4. Magnetic field caused by current element: µµ0 I ~er × d~l . 4π r2 ~ + E), ~ F~ = I~ × Bl. ~ 5. F~ = e(~v × B ~ = dB

where q is charge, which traveled through potential drop U ; work of electromotive force is 6. Simple conclusions of Gauss’ and circulation A = Eq. theorem: 2 2 a) charged wire E ∝ 1/r, direct current B ∝ 8. WC = CU /2, WL = LI /2. 1/r, b) planar current B = Const., charged 9. E = −dΦ/dt = −LdI/dt, Φ = BS. plane E = Const.; 10. Nonlinear elements: graphical method — c) inside a charged sphere and inside an infinite find solution in U -I coordinates as a interseccylindric surface E = 0, if current flows along tion point of a nonlinear curve and a line represcylindrical surface, then also B = 0, enting Ohm/Kirchoff laws. In case of many solud) if charge and current volume density is unitions investigate stability — some of the soluform, the field is ∝ r (∝ x) for sphere, cylinder tions are usually unstable. and layer. 11. Approximate solving: if τobservable ≫ τRC 7. Long solenoid: inside B = Inµµ , out0 or τLR . Quasiequilibrium is formed — either side 0, elsewhere Bk ∝ Ω; Φ = N BS and IC ≈ 0 (wire is ”broken” near C) or EL ≈ 0 (L is L = Φ/I = V n2 µµ0 . short-circuited). If ≪, then according change of 8. Measuring magnetic field with a small coil charge or current is small, ∆Q ≪ Q or ∆I ≪ I; in pulsed operating mode of galvanometer: q = potential of C and flow of current in L are prac- R U R dt = N S∆B/R. tically constant. energy of system of charges: Z 12. If L 6= 0, then I(t) is continuous function. 9. Potential X qi qj 1 = ϕ(~r)dq, dq = ρ(~r)dV. 13. If in a superconducting contour L = Π = rij 2 i>j Const, then according contour current I = Const (more universally: magnetic flux through 10. To find force between surface parts of uniformly charged sphere or cylindrical surface, incontour Φ = Const.). 14. Mutual inductance: magnetic flux through vestigate equivalent pressure force.

b) method of contour currents; c) equivalent cirIn case of that angle the diffraction maximum of cuits (3-terminals: triangle, star; 2-terminal: r a contour Φ = L I + L I (I — current in 1 1 1 12 2 2 the first point overlaps with the first minimum of and E in series). second contour). Theorem: L12 = L21 ≡ M . the second point. 4. Special methods: resistivity of infinite circuit, resistance between neighbour nodes in infinite 6. Bragg theory: if distance between ion planes IX Electromagnetism grid. is a, then x-ray reflects, if incidence angle meets the restriction: 5. Alternating current: same as direct current; 1. F = kq1 q2 /r2 , Π = kq1 q2 /r — Kepler’s 2a sin α = kλ.

I2. Gauss’ theoremI I ~ S ~ = Q, ~ S ~ = 0, ~gdS ~ = 4πM. εε0 Ed Bd

laws are applicable (Ch. XII).

11. In a point located in equal distance of all charges (e.g. inside a sphere or on an axis of cylinder), ϕ = kQ/r. 12. To find the charge (or potential) induced by an electric wire divide a single charge between symmetric positions: problem becomes symmetrical.

13. Conductor shields charges and electric fields, e.g. locations of charges inside a hollow sphere can not be seen from outside (it seems as were the total charge Q located on conductor’s surface)

~ = εε0 E ~ = 26. Fields in substances: D

~ + P~ , where P~ is dielectric polarization vecε0 E ~ = tor (volume density of dipole moment); H ~ ~ ~ ~ B/µµ0 = B/µ0 − J, where J is magnetization vector (volume density of magnetic moment).

14. Capacitances: C = εε0S/d (plane), 27. In an interface between two substances En , 4πεε0 r(sphere), 2εε0 l(ln R/r)−1 (coaxial).

15. Dipole moment: X d~e =

~ qi~ri = ~lq, d~µ = I S.

Dt (= εEt ), Ht (= Bt /µ) and Bn are continuous.

28. Energy density: W = εε0 E 2 + B 2 /µµ0 .

16. Energy and torque of dipole:

~ (B), ~ ~ = d~ × E ~ (B). ~ W = d~ · E M 2 17. Dipole field: ϕ = d~ · ~er /r ; E, B ∝ r−3 . 18. Forces acting on a dipole: F = (E~ d~e )′ , ~ d~µ )′ ; interaction between two dipoles: F = (B F ∝ r4 .

19. Electric and magnetic images: grounded

(superconducting for magnets) planes act as mirrors. Field of a grounded or not grounded sphere can be found as a field of fictive charge inside one or two spheres.

X Thermodynamics

R

19. Heat capacity: Q = c(T )dT . 20. Surface tension: U = Sσ, F = lσ, p = 2σ/R.

6. In case of small ellipticity ε = d/a ≪ 1 trajectory can be considered as a circle, but focuses are shifted.

XI Quantum mechanics

7. Properties of ellipse: l1 + l2 = 2a, α1 = α2 , S = πab.

1. p~ = h¯~k (modulus is h/λ), E = h¯ ω = hν. 2. Interference: as in wave optics. 3. Uncertainty: ∆p∆x ≈ ¯h, ∆E∆t ≈ ¯ h. 4. Spectra: hν = En − Em ; width of line and lifetime: Γτ ≈ h.

and always inside a superconductor B = 0 and thus I = 0 (current flows in surface layer).

23. Charge in homogeneous magnetic field: generalized momentum is constant p′x = mvx + Bye, p′y = mvy − Bxe. Moving along cycloid with average speed v = E/B = F/eB. 24. MHD generator (a — measure along the dir~ ection of E): E = vBa, r = ρa/bc. 25. Hysteresis: S-shaped curve in B-H- or (for coil with core) U -I-coordinates: surface surrounded by it is proportional to thermal loss (energy dissipated in core during one period).

common focuses can only be in the end point of the longer axis.

9. Galilei invariant: ~ε =

~ × ~v L + ~er . γM m

1. pV = m µ RT 5. Oscillator’s (e.g. molecule) natural frequency XIII Theory of relativity 2. Internal energy of one mole U = 2i RT . ν0 : En = (n + 12 )hν0 . If there are many natural P 1. Lorentz transformations p(turns of 43. Volume of one mole at standard conditions is frequencies, then E = i hni νi . dimensional space-time), β = 1/ 1 − v 2 /c2 : 22,4 l. 6. Tunneling effect: barrier Γ with width pl is easx′ = β(x − vt), y ′ = y, t′ = β(t − vx/c2 ) γ γ−1 ily penetrable, if Γτ ≈ h, where τ = l/ Γ/m. 4. pV = Const. (and T V = Const.). p′x = β(px − mv), m′ = β(m − px v/c2 ) 7. Bohr’s model: En ∝ 1/n2. In a circular 5. γ = cp /cv = (i + 2)/i. orbit there are integer number of wavelengths 2. Length of 4-dimensional vector: 6. Boltzman’s distribution: 2 2 2 2 2 2

λ = h/mv. ρ = ρ0 e−µgh/RT = ρ0 e−U/kT . 20. Sphere’s (cylinder’s) polarization in homo- 7. Maxwell’s distribution (how many molecules 8. Compton effect — photon is scattered from 2 electron, ∆λ = λC (1 − cos θ). geneous (electric) field: superposition of two ho- have speed v) ∝ e−mv /2kT . mogeneously charged spheres (cylinders), d ∝ 8. If ∆p ≪ p, then ∆p = ρg∆h. 9. Photoeffect: A + mv2 /2 = hν. I-U -graph: p E. current begins already, if U < 0, near large U 9. p = 13 mn¯v2 , v¯ = 2kT /m, dν = v · dn. 2 21. Momentum of Eddy’ currents: ∆p ∝ B /a, 10. Carnot’s cycle: 2 adiabats, 2 isotherms. η = saturates. where a is characteristic geometrical measure. 10. Stefan-Boltzman: P = σT 4 . (T − T )/T . Derivation in S-T -coordinates.

22. In case of fast processes inside a conductor

8. Contact point of an ellipse and a circle with

1

2

1

11. Inverse cycle: η = T1 /(T1 − T2 ) XII Kepler laws 12. Entropy: dS = dQ/T . 2 13. I law of thermodynamics: δU = δQ + δA 1. F = γM m/r , Π = −γM m/r. 14. II law of thermodynamics: ∆S ≥ 0 (and 2. Gravitational pull of two point masses: tra-

ηreal ≤ ηCarnot ).

15. Gas Zwork (look also p. 10)

i pdV, adiabatic: A = ∆(pV ) 2 P 16. Dalton’s law: p = pi . A=

s =c t −x −y −z

m20 c2 = m2 c2 − p2x − p2y − p2z

3. Adding velocities:

w = (u + v)/(1 + uv/c2 ).

4. Doppler effect: ν ′ = ν0

p (1 − v/c)/(1 + v/c).

5. Space turns: tanh ϕ = v/c; sinh, cosh, tanh instead of sin, cos, tan. Property: cosh2 ϕ − sinh2 ϕ = 1.

6. Shortening of length: l′ = l0 /β. 7. Lengthening of time: t′ = t0 β. jectory of both of them is ellipse, which center is 8. Simultaneity is relative. system’s mass center. 9. F~ = d(m~v )/dt, where m = m0 β. 3. While moving in a central force field, radius 10. Ultrarelativistic approximation: v ≈ c, p p vector covers equal areas in equal times.

4. Two planets’ revolution periods squares are

p ≈ mc,

1 − v 2 /c2 ≈

2(1 − v/c).

17. Boiling: pressure of saturated vapour is p0 . as cubes of longer semiaxis: T12 /T22 = a31 /a32 . 18. Heat flux P = kS∆T /l (k — thermal conductivity factor); analogy to direct current cir- 5. Total energy of a body moving in an elliptical cuits (in correspondence are P and I, ∆T and orbit: U ).

E = −γM m/2a.

Composed by J. Kalda — [email protected], 6204174, 56214406; Translated by U. Visk — [email protected]