Chapter 2: Q110E (page 70)

Equations (2-38) relate momentum and total energy in two frames. Show that they make sense in the non-relativistic limit.

Short Answer

Expert verified

In non-relativistic collisions, the system’s mass does not change, which is shown in the following solution with the help of relations between momentum and energy formula.

Step by step solution

State the relation of energy and momentum in two frames.

The relations between momentum and energy in two frames are expressed as,

$p'_{x} = γ_{v} [p_{x} - \frac{V}{C} (\frac{E}{C})] E' = γ_{v} [E - v p_{x}]$

Express the relativistic momentum expression in the classical limit.

Classically, $\frac{v}{c} < < 1 \Rightarrow γ_v \approx 1 & E ≅ {mc}^{2}$ , the momentum expression becomes,

$p_{x}^{'} = [p_{x} - \frac{v}{c} (\frac{m c^{2}}{c})] p_{x}^{'} = p_{x} - m v$

which is momentum in classical Galilean transformation. Similarly, the energy equation will approximate to,

$E' = E - v p_{x} \frac{E'}{c} = \frac{E}{c} - \frac{v}{c} p_{x} \frac{E'}{c} = \frac{E}{c} m' c = m c ∵ \frac{v}{c} < < 1$

This implies that mass remains the same and is valid for the non-relativistic frame. Therefore, the relativistic momentum and energy equations are valid for classical limits too.