Problems

For this question, you may use the following approximations, valid if \(\theta \) is small: \ \(\sin\theta \approx \theta\) and \(\cos\theta \approx 1-\theta^2/2\,\). A satellite \(X\) is directly above the point \(Y\) on the Earth's surface and can just be seen (on the horizon) from another point \(Z\) on the Earth's surface. The radius of the Earth is \(R\) and the height of the satellite above the Earth is \(h\).

1. Find the distance \(d\) of \(Z\) from \(Y\) along the Earth's surface.
2. If the satellite is in low orbit (so that \(h\) is small compared with \(R\)), show that $$d \approx k(Rh)^{1/2},$$ where \(k\) is to be found.
3. If the satellite is very distant from the Earth (so that \(R\) is small compared with \(h\)), show that $$d\approx aR+b(R^2/h),$$ where \(a\) and \(b\) are to be found.