8.4. Energy Conservation#
The idea of Kinetic Energy was eventually appreciated as a part of a much broader concept. We use the term freely, but it’s a subtle thing and the 17th, 18th, 19th and 20th centuries saw repeated recalibration of the energy-idea. It was not until nearly the middle of the 1800s that heat was carefully studied by many, culminating when Joule did his careful water-mixing experiment. Remember that young man, the eventual Lord Kelvin, who attended that fateful 1847 James Joule lecture? He was about the first person to begin to regularly use the word “energy” around 1850. It’s so overused now. (Tired? You apparently lack “energy.” We have an “energy crisis.” “Energy production” is a common phrase, but incorrectly used. Who do you know who is an “energetic person”?)
Einstein will teach us a lot about energy, but we’ll follow a conventional path until we get to him. One thing that stands the test of time, however, is energy conservation.
Let’s write down two conservation sets of equations between a one-dimensional (no vector symbols required) collision of two objects with initial momenta \(p_0(1), p_0(2)\) and kinetic energies \(K_0(1),K_0(2)\) and final momenta \(p(1), p(2)\) and kinetic energies \(K(1),K(2)\).
Perfectly elastic collision:
An everyday, inelastic collision:
Here \(K(\text{parts}\) accounts for all of the energy lost to internal motion of the parts of the objects. So, if you could capture all of the molecular-level energies of the parts (that became heat), then you could balance energy as well as momentum. So while kinetic energy is not conserved in an everyday, inelastic collision, total energy is conserved.
**Putting on the Brakes **
In normal driving, you acquire a speed and hence gain kinetic energy through the transformation of chemical energy into kinetic energy. Then, in order to stop, you need to remove kinetic energy from your car and for that you “step on the brakes” which means you engage a mechanism in each of your four wheels that forces two plates to rub against one another: one is rotating with the wheel, and the other is fixed to the car. That is, friction is your stopping friend. By now you know that this means that kinetic energy is converted into heat. Brakes can get very hot! So that’s lost energy. In order to speed up again, you’ve got to burn more gasoline.
Hybrid and electric cars are smarter. When a Toyota Prius stops the car reacts to your brake pedal much differently: it causes the motor that normal propels the car forward (converting electrical energy into mechanical energy) to reverse—it becomes a generator (converting mechanical energy back into electrical energy). So when you slow down your kinetic energy is converted by the (now) generator into either electrical charge storage in a capacitor or a current that directly recharges the car’s batteries. This means that 30-50% of the otherwise wasted heat in frictional braking is recovered to help you go farther than you otherwise might go on a battery charge alone. Of course, a Prius is usually a hybrid, so a gasoline engine is sometimes charging the batteries and also propelling the car forward. But an all-electric car, like a Tesla, is totally reliant on batteries and recovered kinetic energy through this “regenerative braking” mechanism.
There is a whole new racing venue called Formula E which are completely electric Formula-1-looking racing cars. As of this writing, there have been three Formula E seasons and a new “Gen2” engineering platform is to be deployed for the 2018-19 season. These cars have maximum power of more than 300 hp (always reported as “250 kW,” as befitting a purely electrical device.) Unlike Gen 1, these new cars will go 45 minutes without replacing batteries. Again, there are two kinds of braking, friction and regenerative. In the new cars, the decision is made by the car’s electronics, whereas in Gen 1 the driver had to decide whether to use regenerative or frictional braking each time.
Are these cars fast? They’re nearly competitive with traditional Formula 1: 180 mph and 0-60 mph in under three seconds. They just sound strange.
With Mercedes, Audi, Porsche, Ford, BMW, Jaguar, Nissan, DS-Citroen, and McLaren (battery development), the technology will spill over into the commercial market as has happened in traditional racing for a century.
This is energy conservation that will change how we all get around some day.
Somewhere in your life, you probably learned that there are many kinds of energy: potential, thermal, chemical, electrical, magnetic, nuclear, gravitational, and elastic. In the above, \(K(\text{parts})\) could represent the loss (as a positive number in that equation—you have to add it back in at the end) of one or more of these kinds of energies.
Wait. Energy is a kind of universal idea, but why so much complication?
Glad you asked. That’s a really good question. You want to try to find something about all of these that’s the same and to say that they can all produce “work” seems unsatisfactory, doesn’t it. Albert Einstein’s tee-shirt equation will bring a lot of this together, so stay tuned. But I appreciate your energy.
There’s one particular “kind” of energy that gets special mention: Potential Energy. Here are some fine, textbook-sounding definitions:
A body or a system has energy if it can do work, that is, move something against a force. Your hammer headed towards a nail has energy. A lightening bolt has energy. A heated, pressurized boiler has energy.
The particular kind of energy associated with motion is Kinetic Energy. That’s the hammer above.
The particular kind of energy associated with position is Potential Energy.
That last one bears some explanation. “Position” means that some object is being held back or prevented from being where it would be without that constraint. The simplest way to think of this is potential energy due to height.