Calculate the velocity of rocket 2 relative to the Earth, using the

(i) Galilean transformation equation.

(ii) relativistic transformation equation.

Comment on your answers in (a).

The rest mass of rocket 1 is 1.0×10³kg. Determine the relativistic kinetic energy of rocket 1, as measured by an observer on Earth.

(i) u'_x= u_x + v = 0.60c + 0.80c = 1.40c;

(ii) \({u_x}^\prime \left( { = \frac{{{u_x} + v}}{{1 + \frac{{{u_x}v}}{{{c^2}}}}}} \right) = \frac{{0.60c + 0.80c}}{{1 + \frac{{0.60c \times 0.80c}}{{{c^2}}}}}\);
\(\left( { = \frac{{1.40c}}{{1.48}}} \right) = 0.95c\);
Award [1] for answers that use v = –0.80 c to get an answer of –0.38 c.

the answer to (a)(i) exceeds c / the answer to (a)(ii) does not exceed c;
hence the Galilean transformation is not valid / the relativistic transformation must be used / OWTTE;

\(\gamma  = \frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }} = \frac{1}{{\sqrt {1 = \frac{{{{0.80}^2}{c^2}}}{{{c^2}}}} }} = 1.7\);
\({E_K} = \left[ {\gamma  - 1} \right]{m_0}{c^2} = \left[ {1.667 - 1} \right] \times 1.0 \times {10^3} \times {\left[ {3.0 \times {{10}^8}} \right]^2}\);
=6.0×10¹⁹J
Award [0] for answers that use E_k = ½ mv² to get an answer of 2.9×10¹⁹ J.