Question

Figure 37-20 shows the triangle of Fig 37-14 for six particles; the slanted lines 2 and 4 have the same length. Rank the particles according to (a) mass, (b) momentum magnitude, and (c) Lorentz factor, greatest first. (d) Identify which two particles have the same total energy. (e) Rank the three lowest-mass particles according to kinetic energy, greatest first.

Short answer

a) $m_3=m_4=m_6>m_1=m_2=m_5$

b) $p_1>p_2=p_3>p_4>p_5=p_6$

c) ${\gamma }_1>{\gamma }_2>{\gamma }_3>{\gamma }_3>{\gamma }_5>{\gamma }_6$

d) Particle 2 and 4 have same length of slanted lines, therefore have same total energy.

e) The particle 1, 2, and 5 have lowest mass as the base length is smaller than other three particles.

Step-by-step solution

Momentum and energy

The momentum and energy relation is given by

$E^2={\left(pc\right)}^2+{\left(m{c}^2\right)}^2$

This equation can be represented in the form of a triangle Fig 37-14 as shown below

It can also be shown that for this triangle

$\sin \theta =\beta \&\cos \theta =\frac{1}{\gamma }$

Mass

An object at rest has some energy called rest mass energy and is expressed as

$E_{rest}=m{c}^2$

In fig 37-20, the base of triangle represents rest mass energy. As rest mass energy is only dependent on mass with$c^2$term being constant. Therefore, the particle with greatest rest energy will have greatest mass, Particles 3,4, and 6 have same base length, therefore same rest energy. And particles 1, 2, and 5 have same mass.

$m_3=m_4=m_6>m_1=m_2=m_5$

Momentum magnitude

The vertical length represents momentum magnitude $p$. The particle 1 has highest height therefore greatest momentum. The particle 2 and 3 has the same height therefore same momentum but less than the particle 1. Particle 5 and 6 have same height therefore same momentum

$p_1>p_2=p_3>p_4>p_5=p_6$

Lorentz factor

The result of 2^nd postulate of the special theory of relativity is that the clocks run slower for a moving object when measured from a rest frame. The factor by which the clock is running differently is called the Lorentz factor.

The expression for Lorentz factor is

$Y=\frac{1}{\sqrt{i-{\beta }^2}}$

Here $\beta$is the speed parameter $\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$c$}\right.$.

We know that

$\begin{array}{l}\cos \theta =4\gamma \\ \gamma =\frac{1}{\cos \theta }\end{array}$

As $$" width="9" height="19" role="math">θincreases $\cos \theta$decreases and as a result $\gamma$will increase. Hence greatest angle $\theta$will have greatest Lorentz factor. Therefore, the Lorentz factor ranking will be

${\gamma }_1>{\gamma }_2>{\gamma }_3>{\gamma }_4>{\gamma }_5>{\gamma }_6$

As for part (d), particle 2 and 4 have same length of slanted lines, therefore have same total energy.

For part (e), the particle 1, 2, and 5 have lowest mass as the base length is smaller than other three particles.