By Russell L. Herman

Advent and ReviewWhat Do i have to understand From Calculus?What i would like From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe easy Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical ideas of ODEsNumericalRead more...

summary: creation and ReviewWhat Do i must be aware of From Calculus?What i want From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe basic Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical strategies of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali

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This is the Newtonian approximation and does not provide enough of an approximation for terrestrial speeds. Thus, we need to expand γ in powers of v/c. First, we rewrite γ as γ= 1 1− v2 c2 = 1− v c 2 −1/2 . Using the binomial expansion for p = −1/2, we have γ ≈ 1+ − 1 2 − v2 c2 = 1+ v2 . 39. Time Dilation Example The average speed of a large commercial jet airliner is about 500 mph. If you flew for an hour (measured from the ground), then how much younger would you introduction and review be than if you had not taken the flight, assuming these reference frames obeyed the postulates of special relativity?

Recall that cosh x = e x + e− x . 2 Hyperbolic functions and trigonometric functions are intimately related: cos(ix ) = cosh x, sin(ix ) = −i sinh x. If we let x = iθ, then we have that cosh(iθ ) = cos θ and cos(ix ) = cosh x. Similarly, we can show that sinh(iθ ) = i sin θ and sin(ix ) = −i sinh x. 8 Binomial Expansion Another series expansion which occurs often in examples and applications is the binomial expansion. This is simply the expansion of the expression ( a + b) p in powers of a and b.

An example of this occurs in statistical mechanics. Statistical mechanics is the branch of physics which explores the thermodynamic behavior of systems containing a large number of particles. An important tool is the partition function, Z. This function is the sum of terms, e− n /kT , over all possible quantum states of the system. Here, n is the energy of the nth state, T the temperature, and k is Boltzmann’s constant. Given Z, one can compute macroscopic quantities, such as the average energy, < E >= − ∂ ln Z , ∂β where β = 1/kT.