I've been given the following problem (admittedly on a homework sheet), which I've solved, but I feel there has to be a neater method of solving it:
A satellite falls freely towards the Earth starting from rest at a distance R, much larger than the Earth’s radius. Treating the Earth as a point of mass M, letting the time taken to reach the Earth be T, the distance of the satellite from the Earth at time t be r(t), with G the gravitational constant, working from the equation of energy conservation for the satellite, prove:
$$\ T = \frac{\pi}{2\sqrt2} (\frac{R^3}{GM})^{1/2}$$
Working through this, I got this $ E = \frac{1}{2}m\dot{r}^2 - \frac{GMm}{r}$, where m is the mass of the satellite, for the equation of energy conservation (assuming no friction). This leads to a rather nasty integral when solving, which does indeed give the correct answer. However, the integral, namely $ \int\sqrt\frac{r}{Er+GMm}dr $ is incredibly annoying and time-consuming to solve, and seems out of character with the neatness of the other problems on the sheet. I was wondering if there's perhaps a better way of doing this, or indeed some clever trick to solve the integral. Any insight would be much appreciated.