We have the same sort of problem tracking energy that we add to the system when I give the block a push. Chemical energy in sugars causes muscle cells to change their shape, contracting muscles that do work on my arm, which exchanges energy with the block via the normal force between block and skin. The chemical energy itself originally came from thermonuclear fusion reactions in the sun, and the free energy released in those interactions can be tracked back to the Big Bang, with a lot of imagination and sloughing over of details. Energy, it turns out, has “always” been around (as far back in time as we can see, literally) but is constantly changing form and generally becoming more disorganized as it does so.

In this textbook, we will say a little more on this later, but this is enough for the moment. We will summarize this discussion by remarking that non-conservative forces, both external (e.g. friction acting on a block) and internal (e.g. friction or collision forces acting between two bodies that are part of “the system” being considered) will often do work that entirely or partially “turns into heat”

– disappears from the total mechanical energy we can easily track. That doesn’t mean that it is has truly disappeared, and more complex treatments or experiments can indeed track and/or measure it, but we just barely learned what mechanical energy is and are not yet ready to try to deal with what happens when it is shared among (say) Avogadro’s number of interacting gas molecules.

3.6: Power

The energy in a given system is not, of course, usually constant in time. Energy is added to a given mass, or taken away, at some rate. We accelerate a car (adding to its mechanical energy). We brake a car (turning its kinetic energy into heat). There are many times when we are given the rate at which energy is added or removed in time, and need to find the total energy added or removed. This rate is called the power.

The units of power are clearly Joules/sec = Watts. Another common unit of power is “Horsepower”, 1 HP = 746 W. Note that the power of a car together with its drag coe cient determine how fast it can go. When energy is being added by the engine at the same rate at which it is being dissipated by drag and friction, the total mechanical energy of the car will remain constant in time.

Example 3.6.1: Rocket Power

A model rocket engine delivers a constant thrust F that pushes the rocket (of approximately constant mass m) up for a time tr before shutting o . Show that the total energy delivered by the rocket engine is equal to the change in mechanical energy the hard way – by solving Newton’s Second Law for the rocket to obtain v(t), using that to find the power P , and integrating the power from 0 to tr to find the total work done by the rocket engine, and comparing this to mgy(tr ) + 12 mv(tr )2, the total mechanical energy of the rocket at time tr .