- The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.
At time \(t\) seconds, where \(t \geqslant 0\)
- \(\quad P\) is a distance \(x\) from \(O\)
- \(P\) is moving with speed \(v\)
- \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)
Air resistance is modelled as being negligible.
- Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\)
Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
- Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
- Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.