Question 1:
difficult
A body of superdense material with mass twice the mass of earth but size very small compared to the size of earth starts from rest from \(h \ll R\) above the earth’s surface. It reaches earth in time \(t\). Then:
The acceleration of the earth towards the body is \(a_E = 2g\) and the body towards the earth is \(a_B = g\). The relative acceleration is \(a_{rel} = 3g\). Using \(h = \frac{1}{2} a_{rel} t^2\), we get \(t = \sqrt{\frac{2h}{3g}}\).