What is relativistic speed

Special theory of relativity

Historical: In his work from 1905, EINSTEIN asks the question in the headline: "Does the inertia (note: mass) of a body depend on its energy content?" You can briefly consider this work. It first deals with a special problem, namely the emission of radiation. In the end, however, EINSTEIN comes to the following generalization:

Hints

  • Following the sizes used by EINSTEIN, the sizes commonly used today are shown in red for your convenience.
  • Erg is an older unit of energy that is rarely used today.
  • EINSTEIN often stated the speed of light in the unit cm / s.
  • In the lower section, EINSTEIN already expresses himself about an experiment with radium salts (these were the substances common at the time that emitted radiation and entered into nuclear reactions) that could serve to test the theory.
  • The short excerpt from EINSTEIN's original work expresses the fact that you used to calculate binding and reaction energies in the intermediate level: \ (\ Delta E = \ Delta m \ cdot {c ^ 2} \)

EINSTEIN's considerations finally led to the fact that he was able to derive a proportionality between the dynamic mass \ (m (v) \) and the relativistic total energy \ (E \). The following applies:

Relativistic total energy

\ [E = m (v) \ cdot {c ^ 2} \]

Here E: the relativistic total energy of a body, m (v): dynamic mass of a body and c: the speed of light in a vacuum

Mass and energy are linked to one another via this fundamental relationship, which is also referred to as the Equivalence of mass and energy. In the article written for a broad, interested readership (link at the end of this article) Einstein explains how the above relationship merges the conservation laws for mass and energy into a single comprehensive conservation law. Integral calculus is a prerequisite for a sustainable derivation of this famous formula, which is why we have dispensed with it at this point.

Resting energy

According to the above relationship, an energy can also be assigned to a body with a velocity of zero, which is called the rest energy E0 designated:
\ [E (v) = m (v) \ cdot {c ^ 2} \ Rightarrow E (v) = \ frac {{{m_0}}} {{\ sqrt {1 - {{\ left ({\ frac { v} {c}} \ right)} ^ 2}}}} \ cdot {c ^ 2} \ mathop \ rm {\; \; and \; for \; \;} v = 0 \; \; \; E (0) = {m_0} \ cdot {c ^ 2} \]
\ [{E_0} = {m_0} \ cdot {c ^ 2} \]

Kinetic energy

The faster a body is moved, the greater its dynamic mass and thus its total energy. The body's kinetic energy is the difference between its total energy and rest energy:
\ [{E_ {kin}} = E (v) - {E_0} \ Rightarrow {E_ {kin}} = m (v) \ cdot {c ^ 2} - {m_0} \ cdot {c ^ 2} \ Rightarrow {E_ {kin}} = \ left ({m (v) - {m_0}} \ right) \ cdot {c ^ 2} \]

Confidence-building measure: Non-relativistic approximation for the kinetic energy

In mathematics one can learn from the series expansion that for small x (ie x << 1) the following applies: \ (\ frac {1} {\ sqrt {1 - x}} \ approx 1 + \ frac {1} {2 } \ cdot x \). This approximation should now be applied to the relativistically correct expression for the kinetic energy, where x is replaced by the quotient of v and c.
\ [{E_ {kin}} = \ frac {{{m_0}}} {{\ sqrt {1 - {{\ left ({\ frac {v} {c}} \ right)} ^ 2}}}} \ cdot {c ^ 2} - {m_0} \ cdot {c ^ 2} \ Rightarrow {E_ {kin}} \ approx {m_0} \ cdot {c ^ 2} \ cdot \ left ({1 + \ frac {1 } {2} \ cdot {{\ left ({\ frac {v} {c}} \ right)} ^ 2}} \ right) - {m_0} \ cdot {c ^ 2} \ Rightarrow {E_ {kin} } \ approx \ frac {1} {2} \ cdot {m_0} \ cdot {v ^ 2} \]
This means that for \ (\ frac {v} {c} \ ll 1 \ Rightarrow v \ ll c \) the relation for the relativistically correctly calculated kinetic energy changes into the well-known formula for the kinetic energy in classical physics.

Indicates a common mistake

Some students believe that they are satisfied with the theory of relativity when calculating the kinetic energy if they use the classical formula \ (E _ {\ text {kin}} = \ frac {1} {2} \ cdot m \ cdot v ^ 2 \) replace the mass with the dynamic mass m (v). As you can easily check, this does not lead to the above correct relationship for the kinetic energy.