General Physics, 2nd Edition
General Physics, 2nd Edition
2nd Edition
ISBN: 9780471522782
Author: Morton M. Sternheim
Publisher: WILEY
Question
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Chapter 25, Problem 26E

(a)

To determine

The fractional error when v=0.1c .

(a)

Expert Solution
Check Mark

Answer to Problem 26E

The fractional error when v=0.1c is 0.0075 .

Explanation of Solution

Write the expression for the nonrelativistic approximation for kinetic energy of the particle,

    Kapprox=12mv2        (I)

Here, Kapprox is the nonrelativistic approximation for kinetic energy of the particle, v is the speed of the particle and m is the mass of the particle.

Write the expression for the relativistic exact kinetic energy of the particle,

    Kexact=mc2(11(v/c)21)        (II)

Here, Kexact is the relativistic exact kinetic energy of the particle, m is the mass of the particle, v is the speed of the particle and c is the speed of the light.

Write the expression for the fractional error,

    ferror=|KexactKapprox|Kexact        (III)

Here, ferror is the fractional error.

Substitute (I) and (II) in (III),

    ferror=|mc2(11(v/c)21)12mv2|mc2(11(v/c)21)        (IV)

Conclusion:

Substitute 0.1c for v in (IV),

    ferror=|mc2(11(0.1c/c)21)12m(0.1c)2|mc2(11(0.1c/c)21)=|mc2(11(0.1)21)(0.005)mc2|mc2(11(0.1)21)=|(0.0050378)(0.005)|(0.0050378)=0.0075

Therefore, the fractional error when v=0.1c is 0.0075 .

(b)

To determine

The fractional error when v=0.5c .

(b)

Expert Solution
Check Mark

Answer to Problem 26E

The fractional error when v=0.5c is 0.192 .

Explanation of Solution

Write the expression for the nonrelativistic approximation for kinetic energy of the particle,

    Kapprox=12mv2        (I)

Here, Kapprox is the nonrelativistic approximation for kinetic energy of the particle, v is the speed of the particle and m is the mass of the particle.

Write the expression for the relativistic exact kinetic energy of the particle,

    Kexact=mc2(11(v/c)21)        (II)

Here, Kexact is the relativistic exact kinetic energy of the particle, m is the mass of the particle, v is the speed of the particle and c is the speed of the light.

Write the expression for the fractional error,

    ferror=|KexactKapprox|Kexact        (III)

Here, ferror is the fractional error.

Substitute (I) and (II) in (III),

    ferror=|mc2(11(v/c)21)12mv2|mc2(11(v/c)21)        (IV)

Conclusion:

Substitute 0.5c for v in (IV),

    ferror=|mc2(11(0.5c/c)21)12m(0.5c)2|mc2(11(0.5c/c)21)=|mc2(11(0.5)21)(0.125)mc2|mc2(11(0.5)21)=|(0.1547)(0.125)|(0.1547)=0.192

Therefore, the fractional error when v=0.5c is 0.192.

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