Assume that there is a constant ratio between stellar mass and light in the disks of spiral galaxies, and that disks have a constant surface brightness. use the scaling with radius r, of the stellar mass within such a radius, and the scaling of circular velocity, vc, with enclosed mass (ignoring dark matter), to "explain" the tully–fisher relation, l ? vc4. the fourth-power dependence is indeed what is observed at infrared wavelengths, at which light traces stellar mass relatively well. in reality, however, dark matter cannot be ignored, and spiral disks do not have a constant surface brightness.

To place a communications satellite into a geosynchronous orbit at an altitude of 22,240 mi above the surface of the earth, the satellite first is released from a space shuttle , which is in a circular orbit at an altitude of 185 mi, and then is propelled by an upper-stage booster to its final altitude. as the satellite passes through a, the booster's motor is fired to insert the satellite into an elliptic transfer orbit. the booster is again fired at b to insert the satellite into a geosynchronous orbit. knowing that the second firing increases the speed of the satellite by 4810ft/s, determine (a.) the speed of the satellite as it approaches b on the elliptic transfer orbit, (b.) the increase in speed resulting from the first firing at a.