Modern Physics
Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Chapter 2, Problem 16P

(a)

To determine

The momentum and energy in S frame.

(a)

Expert Solution
Check Mark

Answer to Problem 16P

The momentum and energy in S frame are p=m[(uv)/(1uv/c2)]1(uv)2/(1uv/c2)2 and E=mc21(uv)2/(1uv/c2)2.

Explanation of Solution

Write the expression for momentum in S frame,

    p=mu(1v2/c2)1/2        (I)

Here, p is the momentum, v is the velocity, m is the mass of the particle and c is the speed of light.

Write the expression for energy in S frame,

    E=mc2(1u2/c2)1/2        (II)

Here, E is the energy, v is the velocity, m is the mass of the particle and c is the speed of light.

Conclusion:

Write the expression for momentum in S frame, by substituting uv1uv/c2 for u

    p=mu1(u/c)2=m[(uv)/(1uv/c2)]1[(uv)/(1uv/c2)]2(1/c2)=m[(uv)/(1uv/c2)]1(uv)2/(1uv/c2)2        (III)

Write the expression for energy in S frame by substituting uv1uv/c2 for u,

    E=mc21(u/c)2=mc21[(uv)/(1uv/c2)]2(1/c2)=mc21(uv)2/(1uv/c2)2        (II)

The momentum and energy in S frame are p=m[(uv)/(1uv/c2)]1(uv)2/(1uv/c2)2 and E=mc21(uv)2/(1uv/c2)2.

(b)

To determine

Show that E2p2c2 is equal to E2p2c2.

(b)

Expert Solution
Check Mark

Answer to Problem 16P

It is proved that E2p2c2 is equal to E2p2c2.

Explanation of Solution

Write the given expression

  E2p2c2        (I)

Here, E is the energy, p is the momentum, and c is the speed of light.

Write the given expression

  E2p2c2        (II)

Here, E is the final energy, p is the final momentum, and c is the speed of light.

Write the expression for momentum in S frame,

    p=mu(1v2/c2)1/2        (III)

Here, p is the momentum, v is the velocity, m is the mass of the particle and c is the speed of light.

Write the expression for energy in S frame,

    E=mc2(1u2/c2)1/2        (IV)

Here, E is the energy, v is the velocity, m is the mass of the particle and c is the speed of light.

Write the expression for momentum in S frame,

    p=m[(uv)/(1uv/c2)]1(uv)2/(1uv/c2)2        (V)

Here, p is the momentum, v is the velocity, m is the mass of the particle and c is the speed of light.

Write the expression for energy in S frame,

    E=mc21(uv)2/(1uv/c2)2        (VI)

Here, E is the energy, v is the velocity, m is the mass of the particle and c is the speed of light.

Conclusion:

Substitute expression (III) and (IV) in expression (I)

  E2p2c2=(mc2(1u2/c2)1/2)2(mu(1v2/c2)1/2)2c2Further solving=m2c4

Substitute expression (V) and (VI) in expression (II)

  E2p2c2=(mc21(uv)2/(1uv/c2)2)2(m[(uv)/(1uv/c2)]1(uv)2/(1uv/c2)2)2c2Further solving=m2c4

Hence it is proved that E2p2c2 is equal to E2p2c2.

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