Suppose you are navigating a spacecraft far from other objects. The mass of the spacecraft is 3.1 × 104 kg (about 31 tons). The rocket engines are shut off, and you're coasting along with a constant velocity of <0, 26, 0> km/s. As you pass the location <5, 4, 0> km you briefly fire side thruster rockets, so that your spacecraft experiences a net force of <4 × 105, 0, 0> N for 24.5 s. The ejected gases have a mass that is small compared to the mass of the spacecraft. You then continue coasting with the rocket engines turned off. Where are you an hour later? (Think about what approximations or simplifying assumptions you made in your analysis. Also think about the choice of system: what are the surroundings that exert external forces on your system?)

You are the coordinator for a program that is going to take place at night in a rectangular amphitheater in the mountains. you will have no access to any electricity, but you must be able to illuminate the entire grounds. you know the intensity of the light from a lantern varies inversely as the square of the distance from the lantern. suppose the intensity is 90 when the distance is 5 m. a. write an equation to model the situation. b. solve for the constant of variation. c. write the equation to model the situation using the constant () of variation. d. you have been given lanterns with 40 light intensity. use your equation to solve for the distance from the lantern. e. you need to illuminate 225 km. how many meters do you need to light? f. how many lanterns will you need?