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8. why do radio telescopes convert radio waves (analog signals) to electrical (digital) signals for analysis? (hint: what did the article say about the advantages of digital signals? )

From center station, a train departs every 30 minutes on the fast line and a train departs every 50 minutes on the state line. if two trains depart from center station at 8: 00 a. m., one on each of the two lines, what is the next time that two trains, one on each line, will depart at the same time?

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.