1.) What distance does the spacecraft observe between mars and earth if earth based measurements show a distance between those objects of 40.0 million miles?
Please see the attached file.
1. As per the principles of special relativity, the distance between Mars and Earth as measured by the observer in the spacecraft will appear shorter as compared to that measured by an observer on the earth (length contraction) as per the relation :
L = L0√1 - v2/c2
where, L0 = length measured by the observer on earth = 40 million miles
L = length measured by the observer on the aircraft = To find
v = velocity ...
This problem is a good example of application of relativistic length contraction, time dilation and velocity transformation.