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**PLEASE ONLY ANSWER 15C AND 15D thumbs down if not answered**

(a) Explain the different reasoning behind the Galilean and the Lorentz transformation equations. [2] (b) Discuss the significance of the use of 4-vectors in Special Relativity, drawing comparison with 3-vectors where appropriate. Detailed mathematics is not necessary. [6] (c) Show that the 4-force of a particle can be expressed as \( \boldsymbol{F}=m_{0} \boldsymbol{A} \). You may assume the expression for 4-acceleration: \[ \boldsymbol{A}=\gamma(u)\left(\gamma(u) \boldsymbol{a}+\frac{d \gamma(u)}{d t} \boldsymbol{u}, i c \frac{d \gamma(u)}{d t}\right) \] Under what conditions would this expression for \( \boldsymbol{F} \) cease to be valid? [6] (d) Consider a particle of rest mass \( m_{0} \) moving with 4-velocity \( \boldsymbol{U} \) being subject to a 4-force \( \boldsymbol{F} \). By considering the scalar product \( \boldsymbol{U} \). \( \boldsymbol{F} \) show that: \[ c^{2} \frac{d m}{d t}=\boldsymbol{u} \cdot \boldsymbol{f}+\frac{c^{2}}{\gamma(u)} \frac{d m_{0}}{d t} \] where \( \boldsymbol{u} \) and \( \boldsymbol{f} \) are the corresponding 3-D velocity and force respectively. [6]

Lorentz transformations are valid for any speed