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A spherical planet has mass \( M \) and radius \( R \).
The planet may be considered to have all its massconcentrated at its centre. A rocket is launched from the surface of the planet, such that the rocket moves radially away from the planet. The rocket engines are stopped, when the rocket is at a height \( R \) above the surface of the planet. The mass of the rocket, afterits engines have been stopped, is \( m \).
(i) Show that, for the rocket to travel from a height \( \mathrm{R} \) to a height \( 2 \mathrm{R} \) above the planet's surface, the change \( \Delta U \) in the magnitude of the gravitational potential energy of the rocket is given by the expression
\[
\Delta \mathrm{U}=\frac{\mathrm{GM} m}{6 \mathrm{R}}
\]
(ii) During the ascent from a height \( R \) to a height \( 2 R \), the speed of the rocket changes from \( 7,600 \mathrm{~m} \mathrm{~s}^{-1} \) to \( 7,320 \mathrm{~m} \mathrm{~s}^{-1} \). Show that, in SI units, the change \( \Delta E_{K} \) in the kinetic energy of the rocket is given by the expression
\[
\Delta \mathrm{E}_{\mathrm{K}}=2 \cdot 09 \times 10^{6} \mathrm{~m}
\]
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