Q:

Average Velocity

A car travels 100 miles in 2 hours, it then completes the return leg of the journey.   How fast must it travel on the return leg to average 100mph over the total journey. 
My thoughts on  this are that it is impossible as if the total average was 100mph then the total time would be 2 hours but that can't be if the first leg took 2 hours. 
Please tell me if I am missing something

A:

Are you missing something?  
You probably are if this question was asked during a course on relativity.  Anyway, this is a physics site and I'm going to make the question a bit more precise on the reference frames in which the measurements might have taken place:
We observe a car travel 100 miles in 2 hours, it then completes the return leg of the journey.  How fast must it travel on the return leg for the driver to have done the full 200 miles in 2 hours? 
The answer starts from the observation that during the first leg the driver will have aged $2\sqrt{1-\frac{v^2}{c^2}}$ hours, with $v/c \approx 50/670616629 \approx 7.5 \ 10^{-8}$. That is a fraction $5.6 \ 10^{-15}$ short of 2 hours.  
So, the second leg the car should travel at a speed $v'$ such that the driver ages $\sqrt{1-\frac{v'^2}{c^2}} \frac{100 mi}{c}= 11 \ 10^{-15}$ hr.  It follows that $v'$ needs to be a fraction $3 \ 10^{-15}$ short of the speed of light.