The speed of light is, by definition (we define this and derive a definition of distance from there nowadays), c=299792458m/s. We want to know, at this speed, how much distance the radio signals travel in 100 years. Since each year has 365 days (not a leap one though), each day has 24 hours, each hour has 60 minutes and each minute has 60 seconds, the number of seconds in a year will be (365)(24)(60)(60)=31536000, so the distance traveled by the waves in 100 years will be:
Assuming no external forces acting during the collision, total momentum must be conserved, as follows:
The initial momentum can be expressed as follows (taking as positive the initial direction of the ball):
The final momentum can be expressed as follows (since we know that v'b is opposite to the initial vb):
If we assume that Mc >> mb, we can assume that the car doesn't change its speed at all as a result of the collision, so we can replace V'c by Vc in (3).
So, we can write again (3) as follows:
Replacing (2) and (4) in (1), we get:
Simplifying, and rearranging, we can solve for v'b, as follows:
, which is reasonable, because everything happens as if the ball had hit a wall, and the ball simply had inverted its speed after the collision.
The aircraft moves within the air mass that it is part of and in order to compute the ground speed you would add the speed of the tailwind to the airspeed of the aircraft