Introduction to Classical Mechanics · Problem 12.23

4-vectors — Morin

Physics — Mechanics · transformations-2d-motion

Problem

Transformations for 2-D motion ** A particle has velocity

(u'_x, u'_y)

in frame

S'

, which travels at speed

v

in the

x

direction relative to frame

S

. Use the velocity-addition formulas in Section 11.5.2 (Eqs. (11.36) and (11.38)) to show that

\gamma_u = \gamma_{u'}\gamma_v(1 + u'_x v), \quad \text{where } u = \sqrt{u_x^2 + u_y^2} \text{ and } u' = \sqrt{u_x'^2 + u_y'^2} \tag{12.90}

are the speeds in the two frames. Then verify that

E

and

p_x

transform according to Eqs. (12.27), and also that

p_y = p'_y