Challenging +1.2 This is a standard M3 inverse-square law gravitation problem requiring energy conservation with variable force. While it involves integration and algebraic manipulation beyond basic mechanics, the approach is methodical: set up work-energy equation, integrate F·dx from initial to final position, and solve the resulting equation. The given force formula and speeds guide students through a well-established technique, making it moderately above average difficulty but not requiring novel insight.
3. A particle \(P\) of mass \(m\) moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\). When \(P\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). When \(P\) is at a distance \(2 R\) from the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 3 } }\).
Assuming that air resistance can be ignored, find the distance of \(P\) from the surface of the Earth when the speed of \(P\) is \(2 \sqrt { \frac { g R } { 3 } }\).
3. A particle $P$ of mass $m$ moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius $R$. When $P$ is at a distance $x , x \geqslant R$, from the centre of the Earth, the force exerted by the Earth on $P$ is directed towards the centre of the Earth and has magnitude $\frac { m g R ^ { 2 } } { x ^ { 2 } }$. When $P$ is at a distance $2 R$ from the surface of the Earth, the speed of $P$ is $\sqrt { \frac { g R } { 3 } }$.
Assuming that air resistance can be ignored, find the distance of $P$ from the surface of the Earth when the speed of $P$ is $2 \sqrt { \frac { g R } { 3 } }$.
\hfill \mbox{\textit{Edexcel M3 2018 Q3 [7]}}