2) Consider the Lorentz transformation that maps (x,t) into (x',t'), where t and x are both
in time units (so x is really x/c) and all speeds are in c units also.
a) Show that the inverse of the Lorentz transformation at v is the same as the Lorentz
transformation at -v. Why is that required?
b) Show that x^2 - t^2 = x'^2 - t'^2, i.e., that x^2 - t^2 is invariant under Lorentz transformation
A mechanical wave is given by the equation: y(x,t) = 0.5 cos
(62.8x – 15.7t) , Find: (1) Amplitude, frequency, wavelength? (2)
The velocity of the wave? (3) The maximum velocity of the
vibrations? (4) Write down the equation in the opposite
direction?
Consider the following one-dimensional partial differentiation
wave equation. Produce the solution u(x, t) of this equation. 4Uxx
= Utt 0 < x 0 Boundary Conditions: u (0, t) = u (2π, t) = 0,
Initial Conditions a shown below: consider g(x)= 0 in both
cases.
(a) u (x, 0) = f(x) = 3sin 2x +3 sin7x , 0 < x <2π
(b) u (x, 0) = x +2, 0 < x <2π
1. What is the highest value of x that satisfies this equation x(x+4) = -3
A. -1
B. 0
C. 1
D. -3
2. If x2 - 9x = -18, what are the possible values of x?
A. -3 and -6
B. -3 and 6
C. 3 and -6
D. 3 and 6
3. What polynomial can be added to 2x2 - 2x+6 so that the sum is 3x2+ 7x?
A. 5x2+ 5x+ 6
B. 4x2+ 5x+ 6
C....
Consider the following wave equation for u(t, x) with boundary
and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0.
∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1)
∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2)
u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2.
(a) Compute the coefficients a0, a1, a2, ....