The average internal kinetic energy of an object refers to its temperature. If the kinetic energy increases, the temperature of the object increases and vice versa. If that's right, then falling water should heat up? When water falls from a height, its molecules, of course, have some amount of kinetic energy already. So when it starts to fall and its potential energy gets converted to kinetic energy, shouldn't the molecules' kinetic energy also increase? If it does, then its temperature should go up, right? But it doesn't seem like that happens. Can someone explain this please?
It's mean kinetic energy of a body's particles in the frame of reference of the body's centre of mass that is correlated with its temperature. This is the frame in which the body's centre of mass is at rest, so random motion of particles will increase the total energy in this frame, since the total energy is the scalar sum of the particles' energy. This is consistent with the particles' 'thermal' motion being random. The kinetic energy of a falling body is increasing in our frame of reference (at rest with respect to the Earth) but not in the body's (centre of mass) frame.
It is said that the Physicist James Joule during his honeymoon in the Alps in 1849 attempted to verify that the temperature of water at the bottom of a waterfall was higher than at the top.
There is an increase in temperature but it is very small and note that the motion of the water has to be randomised by it hitting something at the bottom of its fall.
If you research this topic you will most probably come across the term mechanical equivalent of heat which heralds a time when the connection between work done and energy (heat) was a matter of debate.