Einstein 1938: The Roller-Coaster

Reference: Evolution of Physics

This paper presents Chapter I, section 7 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.


The Switchback (Roller-Coaster)

Let us trace the motion of that popular thrill-giver, the switchback. A small car is lifted or driven to the highest point of the track. When set free it starts rolling down under the force of gravity, and then goes up and down along a fantastically curved line, giving the occupants a thrill by the sudden changes in velocity. Every switchback has its highest point, that from which it starts. Never again, throughout the whole course of the motion, will it reach the same height. A complete description of the motion would be very complicated. On the one hand is the mechanical side of the problem, the changes of velocity and position in time. On the other there is friction and therefore the creation of heat, on the rail and in the wheels. The only significant reason for dividing the physical process into these two aspects is to make possible the use of the concepts previously discussed. The division leads to an idealized experiment, for a physical process in which only the mechanical aspect appears can be only imagined but never realized.

In a roller-coaster, heat due to friction accompanies mechanical effects.

For the idealized experiment we may imagine that someone has learned to eliminate entirely the friction which always accompanies motion. He decides to apply his discovery to the construction of a switchback, and must find out for himself how to build one. The car is to run up and down, with its starting-point, say, at one hundred feet above ground level. He soon discovers by trial and error that he must follow a very simple rule: he may build his track in whatever path he pleases so long as no point is higher than the starting-point. If the car is to proceed freely to the end of the course, its height may attain a hundred feet as many times as he likes, but never exceed it. The initial height can never be reached by a car on an actual track because of friction, but our hypothetical engineer need not consider that.

To continue moving, the car in a roller-coaster can never exceed the initial height.

Let us follow the motion of the idealized car on the idealized switchback as it begins to roll downward from the starting-point. As it moves its distance from the ground diminishes, but its speed increases. This sentence at first sight may remind us of one from a language lesson: “I have no pencil, but you have six oranges.” It is not so stupid, however. There is no connection between my having no pencil and your having six oranges, but there is a very real correlation between the distance of the car from the ground and its speed. We can calculate the speed of the car at any moment if we know how high it happens to be above the ground, but we omit this point here because of its quantitative character which can best be expressed by mathematical formulae.

There is a definite relationship between distance of the car from the ground and its speed.

At its highest point the car has zero velocity and is one hundred feet from the ground. At the lowest possible point it is no distance from the ground, and has its greatest velocity. These facts may be expressed in other terms. At its highest point the car has potential energy but no kinetic energy or energy of motion. At its lowest point it has the greatest kinetic energy and no potential energy whatever. At all intermediate positions, where there is some velocity and some elevation, it has both kinetic and potential energy. The potential energy increases with the elevation, while the kinetic energy becomes greater as the velocity increases. The principles of mechanics suffice to explain the motion. Two expressions for energy occur in the mathematical description, each of which changes, although the sum does not vary. It is thus possible to introduce mathematically and rigorously the concepts of potential energy, depending on position, and kinetic energy, depending on velocity. The introduction of the two names is, of course, arbitrary and justified only by convenience. The sum of the two quantities remains unchanged, and is called a constant of the motion. The total energy, kinetic plus potential, can be compared, for example, with money kept intact as to amount but changed continually from, one currency to another, say from dollars to pounds and back again, according to a well-defined rate of exchange.

Two mathematical concepts are introduced here: the concepts of potential energy, depending on position, and kinetic energy, depending on velocity. They convert back and forth into each other.

In the real switchback, where friction prevents the car from again reaching as high a point as that from which it started, there is still a continuous change between kinetic and potential energy. Here, however, the sum does not remain constant, but grows smaller. Now one important and courageous step more is needed to relate the mechanical and heat aspects of motion. The wealth of consequences and generalizations from this step will be seen later.

The motion, however, encounters friction, which prevents all of kinetic energy from converting back to potential energy. This friction produces heat. This means that some kinetic energy converts into heat.

Something more than kinetic and potential energies is now involved, namely, the heat created by friction. Does this heat correspond to the diminution in mechanical energy, that is kinetic and potential energy? A new guess is imminent. If heat may be regarded as a form of energy, perhaps the sum of all three heat, kinetic and potential energies remains constant. Not heat alone, but heat and other forms of energy taken together are, like a substance, indestructible. It is as if a man must pay himself a commission in francs for changing dollars to pounds, the commission money also being saved so that the sum of dollars, pounds, and francs is a fixed amount according to some definite exchange rate.

Energy is effect produced by substance. In fact, it is an extension of substance.

The progress of science has destroyed the older concept of heat as a substance. We try to create a new substance, energy, with heat as one of its forms.

Heat is not a substance. It is an effect produced by substance.



Heat as energy is an effect produced by substance. Motion as kinetic energy is also an effect produced by substance. Similarly, the height above the surface of earth as potential energy is also an effect produced by substance.

Energy comes into being as effect of substance. Without substance there cannot be energy. Energy always has substance as its basis. There is no such thing as “pure energy”. 

Both substance and energy are perceived because of the force involved. Therefore, force is the common denominator of substance and energy.


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