A small body starts from rest towards the Sun from the position of Earth, find the total time it takes for the body to crash into the Sun (assume Sun is a point object and no other forces act on the system).
My approach to this question:
- $a = \frac{GM}{r^2}$ for the body at any r distance form Sun.
- Next I found out velocity, $v = \sqrt{2GM(\frac{1}{r} - \frac{1}{R})} $, where R is initial distance
- From here we get $\int_{R}^{0} \frac{dx}{v}$
Now how do I integrate this monstrosity? Is there any easier and faster way than the one above? Any help would be appreciated.