An undiscovered planet, many lightyears from earth, has one moon in a periodic orbit. this moon takes 2060 × 103 seconds (about 24 days) on average to complete one nearly circular revolution around the unnamed planet. if the distance from the center of the moon to the surface of the planet is 235.0 × 106 m and the planet has a radius of 3.90 × 106 m, calculate the moon's radial acceleration .

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.

The speed of light in a material is 0.50 c. what is the critical angle of a light ray at the interface between the material and a vacuum?

Jasmine is diving off a 3-meter springboard. her height in meters above the water when she is x meters horizontally from the end of the board is approximated by the equation h=-x^2+3x+3. what is the maximum height jasmine will reach on her dive?