Tion of light ns equation figure 13.4 and the small-angle approximation imply that de- (equation 13.12, where δ~4gm/尼 4gwb (equation 13.11) and b~ de (from the diagram). the summary claims that together these imply that for the angle θ of images formed by a gravitational lens. exercise 13.6.1. verify that this is true. exercise 13.6.2. the solutions to this equation are θ±-(β± the θ+ solution is clearly outside the einstein ring angle θε. show that meaning that the θ solution is inside the ring. (hint: argue that so, and use this information in equation 13.13, above.) s0, and then

Your starship, the aimless wanderer, lands on the mysterious planet mongo. as chief scientist-engineer, you make the following measurements: a 2.50-kg stone thrown upward from the ground at 14.0 m/s returns to the ground in 5.70 s ; the circumference of mongo at the equator is 2.20×10^5 km ; and there is no appreciable atmosphere on mongo. (a) the starship commander, captain confusion, asks for the following information: what is the mass of mongo? (b) if the aimless wanderer goes into a circular orbit 2.20×10^4 km above the surface of mongo, how many hours will it take the ship to complete one orbit? (c) an unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 44.0 km . to the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 23.0 m/s . if nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?