Proof that the kinetic energy of a moving body is half mv square​

Consider an ideal gas at 27.0 degrees celsius and 1.00 atmosphere pressure. imagine the molecules to be uniformly spaced, with each molecule at the center of a small cube. what is the length l of an edge of each small cube if adjacent cubes touch but don't overlap?

Three resistors are connected in series across a 15-v power supply. if the potential drops across resistors 1 and 2 are 4.1 volts and 3.1 volts, what is the exact potential drop (in volts) across resistor 3?

Arope has one end tied to a vertical support. you hold the other end so that the rope is horizontal. if you move the end of the rope back and forth with a frequency of 4 hz, the transverse wave you produce has a wavelength of 0.5 m. what is the speed of the wave in the rope?