$(1)$ A mad scientist experimenting with quantum gravity has unfortunately created a rip in the fabric of space$-$time, causing the Earth to suddenly have zero velocity. How long will it take for the Earth to hit the surface of the Sun, and with what speed?

Hints: Ignore the motion of the Sun and assume it is fixed in space. Use conservation of mechanical energy to find the velocity of the Earth as it falls to the Sun. Don$$t forget to choose the correct sign for the velocity $($decreasing radius$).$ Set up an integral for the travel time. Non$-$ dimensionalize it by defining a new variable $$ r$/$r$0$ where r$0$ is the initial radial separation of the Earth and Sun $($that is$,1$ au$).$ One of the limits of integration will be really, really small, so you can approximate it as zero. If you set it up right, you will get a constant term in front of an integral that only has $.$ The constant term should have units of time, so rewrite it as some number times the orbital time orb $=2$pr$03/$GM$.$ The integral can be solved analytically with the trigonometric substitution $=$ sin$2$ $,$ but it is okay just to use Wolfram Alpha to do the integral for you. Be sure to report formulas for your answers as well as numerically evaluate the answers.