Kinetic energy

Kinetic energy is energy that a body has as a result of its speed.

left( rac{1}{sqrt{1- v^2/c^2 }} - 1 ight) m c^2

where:

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:Ek is the kinetic energy of the body

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:v is the velocity of the body

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:m is its rest mass

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:c is the speed of light in a vacuum.

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:γmc2 is the total energy of the body

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:mc2 is the rest mass energy (90 petajoule/kg)

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It is an edifying exercise to show that the ratio of this relativistic kinetic energy to the Newtonian kinetic energy given by (1/2)mv2 approaches 1 as v approaches 0, i.e.,

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:lim_{v o 0}{left( rac{1}{sqrt{1- v^2/c^2 }} - 1 ight) m c^2 over mv^2/2}=1.

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This can be done by the techniques of first-year calculus.

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Relativity theory states that the kinetic energy of an object grows towards infinity as its velocity approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary.

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Where gravity is weak, and objects move at much slower velocities than light (e.g. in everyday phenomena on Earth), Newton's formula is an excellent approximation of relativistic kinetic energy.

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The next term in the approximation is 0.375 mv4/c², e.g. for a speed of 10 km/s this is 0.04 J/kg, for a speed of 100 km/s it is 40 J/kg, etc.

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The exact Taylor series is

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:E_k = {1over 2}mv^2 + {3over 8}mleft({v^4over c^2} ight) + {5over 16}mleft({v^6over c^4} ight) + dots ,

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See also

• Recoil
• Joule
• Kinetic projectile

References

• {{Book reference | Author=Serway, Raymond A.; Jewett, John W. | Title=Physics for Scientists and Engineers (6th ed.) | Publisher=Brooks/Cole | Year=2004 | ID=ISBN 0534408427}}
• {{Book reference | Author=Tipler, Paul | Title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.) | Publisher=W. H. Freeman | Year=2004 | ID=ISBN 0716708094}}
• {{Book reference | Author=Tipler, Paul; Llewellyn, Ralph | Title=Modern Physics (4th ed.) | Publisher=W. H. Freeman | Year=2002 | ID=ISBN 0716743450}}

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