Energy is a very fundamental concept that plays a central role
in elementary particle physics. There is one law of physics that
needs to be understood, and that is the relation between energy
and speed of a mechanical object. Here we shall discuss this
law for the case of objects moving with a speed small compared to
the speed of light so that relativistic effects may be ignored. We
are talking about that type of energy, kinetic energy, that you
may have learned about in high school.b The relation between
energy and speed is quadratic: if you accelerate a car to a speed
of 100 km/h then you need (ignoring friction) four times as
much energy (four times the amount of gas) as for accelerating to
50 km/h. Also the converse is true: to bring a car with a speed of
100 km/h to a standstill you need four times as much braking
distance as halting a car going at 50 km/h.
Furthermore the amount of energy needed is proportional to
the mass of the vehicle. To accelerate a car of 2000 kg to some
speed you need twice the energy needed to bring a car of 1000 kg
to that same speed. That is sort of obvious, because you could see
a car of 2000 kg as two cars of 1000 kg tied together.
The considerations above refer to vehicles moving on earth,
but they are more generally valid. To bring a car to a speed of
50 km/h on the moon or on Mars would require the same amount
of energy as on earth. The mass of a car, element in the calculation,
has nothing to do with gravitation. Nonetheless mass is
usually measured by means of weighing the object. Since the
weight of an object is proportional to its mass that works fine
as long as this measurement is always done on the same planet.
But if the weight of an object is measured on the moon it will
be much lighter than on Earth. Yet its mass, used in the energy
calculation, is the same. Thus the measurement of a mass of an
object requires the measurement of its weight and in addition
there is the conversion factor from weight to mass, different in
different gravitational environments.
What is called mass, especially for elementary
particles, has in the first instance nothing to do with weight. It is
the factor that enters in the calculation if the energy must be
computed given the velocity of the object. If you want to have an
idea of a mass measuring machine think of the following. Take
the object of which the mass is to be measured. Bring it up to
some given speed, and shoot it at a plate fixed on a spring. The
plate will be pushed in. The amount by which it is pushed in is
a measure of the mass of the object. This mass-meter would work
equally well on Earth, the moon or Mars, in fact even on a vessel
in empty space.
The important thing is that if the mass of a car is known, then
the amount of energy needed to bring it to some speed can be
calculated. For relativistic speeds the calculation becomes a little
bit more complicated, but the principle remains the same. For a
given body the energy can be computed from its mass and the
velocity by which it moves. That is true for mechanical objects
and it is also true for freely moving particles. Conversely, if the
energy and velocity of a particle are known then its mass can be
computed. Sometimes one knows the speed of a particle and its
energy and in this way its mass can be determined. For example, if
for a given car it is known how much gas has been used to get to
a certain speed it is possible to compute how many people are
seated inside that car (provided the mass of the car itself and the
average weight of the passengers is known). This is essentially the
method by which the mass of a particle with a very small lifetime
can be measured. Measure the energy and the velocity and then
the mass may be determined.
When the velocity of some material object becomes close to the
speed of light things are different from the way described above,
and one must take into account Einstein’s theory of relativity. In
this theory the velocity of light starts playing the role of infinite
speed in the old theory. Thus it is not possible to achieve a speed
exceeding that of light, and when a material body has a velocity
close to the speed of light its energy becomes very large, in fact
infinite in the limit of attaining the speed of light. Velocity
becomes a poor way of describing the state of motion of an object.
In particle physics one almost always works with speeds close to
that of light, and a few numbers will make it clear that using
velocity becomes very awkward.
A typical cyclotron of the fifties accelerated protons to an
energy of 1 GeV (never mind the units at this point). Taking the
velocity of light to be 300 000 km/s this implies a velocity of
212 000 km/s for the protons coming out of this machine. In 1960
the first large CERN machine, the PS, accelerated protons to
30 GeV, implying a velocity of 295 000 km/s. The latest CERN
machine, LEP, accelerated electrons to an energy of 100 GeV,
implying a velocity of 299 999.6 km/s.
A better suited quantity is the momentum. At low speeds
momentum and speed are essentially the same (momentum is
simply mass times the speed, p = mv), and if the speed becomes
twice as large so does the momentum. However, at speeds close to
the speed of light the relationship changes, and the momentum is
very near to the energy divided by the speed of light.
To us the important quantity is the amount of energy (or
momentum) a particle carries, not its speed. To obtain the correct
relation between energy and momentum one must take the massenergy
(that is the energy corresponding to its mass) into account;
even for an object at rest (meaning zero momentum) the energy is
not zero, but according to Einstein it is equal to mc^2, where c
denotes the speed of light.
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