A10 g projectile is traveling east at 2.0 m/s when it suddenly explodes into three pieces. a 3.0 g fragment is shot due west at 10 m/s while another 3.0 g fragment travels 40◦ north east at 12 m/s. what are the speed and direction of the third fragment?

Consider a double atwood machine constructed as follows: a mass 4m is suspended from a string that passes over a massless pulley on frictionless bearings. the other end of this string supports a second similar pulley, over which passes a second string supporting a mass of 3m at one end and m at the other. using two suitable generalized coordinates, set up the lagrangian and use the lagrange equations to find the acceleration of the mass 4m when the system is released. explain why the top pulley rotates even though it carries equal weights on each side.