It is claimed in the lecture that speed c is equal to p /. Show that this last e

Question

It is claimed in the lecture that speed c is equal to p /. Show that this last expression does indeed have units of velocity.

For the remainder of this assignment you may use Maple however you see fit. However please write all answers down by hand (except for parts (7) and (8), which should be handed in as a Maple file). Each of the problems below describe an infinite spring with uniform strength and uniform linear density . Also given is an initial shape u(x, 0) = f(x) of the spring and the initial velocity ut(x, 0) = g(x) of the spring. Let u = u(x, t) denote the resulting longitudinal displacement of the spring.

1. Write a wave equation satisfied by u.

2. Compute the speed c at which a wave would propagate through the spring.

3. Solve for u(x, t) given these initial conditions. Identify R(x) and L(x), the right and left moving wave shapes comprising u(x, t).

4. Verify that u(x, t) solves the wave equation.

5. Compute the total energy of the entire spring. (If your integral is not convenient to solve exactly, you may use Maple’s int command to find a numerical approximation.)

6. Is total energy finite? Is it conserved?

7. Use the plotSTcurves command to plot R(x ct), L(x + ct), and u(x, t) in the same window. Use x0 = 4, x1 = 4, t0 = 0, t1 = 5, numFrames = 60.

8. Use the animateMassiveSpring command to plot u(x, t) as the longitudinal motion of the spring. Use l = .2, h = .5.

1. = 1 = 1 f(x) = sin(2x) /2(x2+1) g(x) = 0

2. = 4 = 1 f(x) = 0 g(x) = xe^x^ 2

3. = 3 = 3 f(x) = 1 /2 (sin x) g(x) = 1 /2 (cos x)

4. = 1 = 1 f(x) = 1 /2 ((x sin(x)) / x2+1 ) g(x) = 0

Explanation / Answer

We know that c is the speed of light

c=/

= v/c is the speed in units of the speed of light.
dt/d relates the time between events on a worldline to the proper time, for a particle of
speed v.

dt'/dt relates the time between events on a worldline for two reference frames
of relative velocity v, with u the particle velocity in the unprimed frame.

If two particles
have velocities u, v in some reference frame then (w) is the Lorentz factor for their
relative velocity.

Since (w) is the Lorentz factor for their relative velocity hence the expression will have unit of velocity.While is the wavelength.

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