While we usually assume that the force of gravity is constant, it actually decreases with altitude according to the formula Fgrav = mgr^2_o / r^2, where ro is the radius of Earth, and r is the distance from the Earth's center.
a) Why can we approximate the force of gravity as constant at the surface of Earth?
b) Show that a(r)dr = vdv.
c) If the particle is fired vertically with velocity vo from Earth's surface, determine its velocity as a function of position r. (Hint: use the result of part b)

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?