Kinetic Energy and the Ultimate Speed Limit
Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is
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Relativistically, at rest we have rest energy The work increases this to the total energy Thus,
11.64 Relativistically, we have .
Relativistic Kinetic Energy
Relativistic kinetic energy is
11.65 When motionless, we have and
11.66 so that at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for at low velocities gives
11.67 A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small velocity here, most terms are very small. Thus the expression derived for here is not exact, but it is a very accurate approximation. Thus, at low velocities,
11.68 Entering this into the expression for relativistic kinetic energy gives
11.69 So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when .
It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that becomes infinite as approaches so that KErel also becomes infinite as the velocity approaches the speed of light. (See Figure 11.22.) An infinite amount of work, and, hence, an infinite amount of energy input, is required to accelerate a mass to the speed of light.
The Speed of Light
No object with mass can attain the speed of light.
So the speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than always add to less than Both the relativistic form for kinetic energy and the ultimate speed limit being have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach—not reach—the speed of light.
Example 11.8 Comparing Kinetic Energy: Relativistic Energy Versus Classical Kinetic Energy
An electron has a velocity . (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is .)
Strategy
The expression for relativistic kinetic energy is always correct, but for (a) it must be used since the velocity is highly relativistic (close to ) First, we will calculate the relativistic factor and then use it to determine the relativistic kinetic energy. For (b), we will calculate the classical kinetic energy, which would be close to the relativistic value if were less than a few percent of and see that it is not the same.
Solution for (a)
- Identify the knowns. ;
- Identify the unknown.
- Choose the appropriate equation.
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- Plug the knowns into the equation.
First calculate . We will carry extra digits because this is an intermediate calculation.
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Next, we use this value to calculate the kinetic energy.
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- Convert units.
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Solution for (b)
- List the knowns. ;
- List the unknown.
- Choose the appropriate equation.
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- Plug the knowns into the equation.
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- Convert units.
11.76
Discussion
As might be expected, since the velocity is 99.0 percent of the speed of light, the classical kinetic energy is significantly off from the correct relativistic value. Note also that the classical value is much smaller than the relativistic value. In fact, here. This is some indication of how difficult it is to get a mass moving close to the speed of light. Much more energy is required than predicted classically. Some people interpret this extra energy as going into increasing the mass of the system, but, as discussed in Relativistic Momentum, this cannot be verified unambiguously. What is certain is that ever-increasing amounts of energy are needed to get the velocity of a mass a little closer to that of light. An energy of 3 MeV is a very small amount for an electron, and it can be achieved with present-day particle accelerators. SLAC, for example, can accelerate electrons to over .
Is there any point in getting a little closer to c than 99.0 percent or 99.9 percent? The answer is yes. We learn a great deal by doing this. The energy that goes into a high-velocity mass can be converted to any other form, including into entirely new masses. (See Figure 11.23.) Most of what we know about the substructure of matter and the collection of exotic short-lived particles in nature has been learned this way. Particles are accelerated to extremely relativistic energies and made to collide with other particles, producing totally new species of particles. Patterns in the characteristics of these previously unknown particles hint at a basic substructure for all matter. These particles and some of their characteristics will be covered in Particle Physics.