Chapter 2: Q97E (page 68)

Show that $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ follows from expressions (2-22) and (2-24) for momentum and energy in terms of m and u.

Short Answer

Expert verified

The relation of relativistic energy in terms of mass and momentum is derived bysquaring the relativistic energy relation in terms of mass and velocity andexpanding it using the binomial identity.

Step by step solution

Square the relativistic energy relation and expand the expression.

The relativistic energy and momentum expressions are given below,

$E = γ_{u} m c^{2}$

Squaring the energy expression and solving further,

$\begin{matrix} E^{2} = \frac{m^{2} c^{4}}{(1 - \frac{u^{2}}{c^{2}})} \\ = m^{2} c^{4} {(1 - \frac{u^{2}}{c^{2}})}^{- 1} \end{matrix}$

Using the binomial expression: ${(1 - x)}^{- 1} = 1 + x + x^{2} + x^{3} + ...$

$\begin{matrix} E^{2} = m^{2} c^{4} [1 + \frac{u^{2}}{c^{2}} + \frac{u^{4}}{c^{4}} + ...] \\ = m^{2} c^{4} [1 + \frac{u^{2}}{c^{2}} (1 + \frac{u^{2}}{c^{2}} + \frac{u^{4}}{c^{4}} + ...)] \\ = m^{2} c^{4} [1 + \frac{u^{2}}{c^{2}} {(1 - \frac{u^{2}}{c^{2}})}^{- 1}] \\ = m^{2} c^{4} + \frac{m^{2} u^{2} c^{2}}{(1 - \frac{u^{2}}{c^{2}})} \end{matrix}$ … (1)

Express the above equation in terms of mass and momentum

The relativistic momentum expression is given below,

$\begin{array}{l} p = γ_{u} m u \\ = \frac{m u}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} \end{array}$

Therefore, the equation (1) becomes,

$E^{2} = m^{2} c^{4} + {(\frac{m u}{\sqrt{1 - \frac{u^{2}}{c^{2}}}})}^{2} c^{2}$

$E^{2} = m^{2} c^{4} + p^{2} c^{2}$

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Show that $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ follows from expressions (2-22) and (2-24) for momentum and energy in terms of m and u.

Short Answer

Step by step solution

Square the relativistic energy relation and expand the expression.

Express the above equation in terms of mass and momentum

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Show thatE2 = p2c2 + m2c4follows from expressions (2-22) and (2-24) for momentum and energy in terms of m and u.

Short Answer

Step by step solution

Square the relativistic energy relation and expand the expression.

Express the above equation in terms of mass and momentum

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

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Show that $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ follows from expressions (2-22) and (2-24) for momentum and energy in terms of m and u.