Q:

Can a cop give a ticket if he's going the other direction? 

Recently a high shool friend of mine got nailed with a $450 speeding ticket on a back country road.  He'll also have to get pay for the class to get 3 (not sure, might be 4) which is 4 hours on two Saturdays.  Overall, with insurance price increasing, his little ticket will probably cost around $1,200 total. 
Now when he got the ticket the cop was going a fair speed (40ish) in the other direction (2 lane road).  Physically the rate of change of speeds between them is greatly increased as the cop was going the other direction.  My friend believes he was going less than the cop said, and his younger brother was watching he speedometer and said the same. 
Are the circumstances of him being pulled grounds to fight the ticket (with a decent chance of winning)?  Can regular radar guns that a cop might use account for the difference in the rate of chamge of speed?  The cop said he was going 32 over and that just sounds plain wrong.  Thanks in advance. 

A:

Moving radar gun readings are held to be admissible evidence, if used properly, and are certified to be accurate within 1 MPH.  Reportedly, they are not accurate if the angle is more than 11 degrees (or something like that); in addition, there are distance limits (too far and the reading is prone to error); and it needs to be calibrated.  Police training records can be subpoenaed to argue that the officer was not properly trained. 
Low-tech speed determinations are admissible, provided the officer is following at a consistent distance (obviously going the same direction).  Another is the actual time taken to pass two highway marks.  This could be unreliable when the officer's vehicle is moving towards the suspect's vehicle (one's perception of when a person has "passed" a distant point depends on your angle relative to the fixed point, and that angle is changing).  
In the case of US v. Sowards, the defendant was pulled over for allegedly speeding (and drugs were then discovered), so the question was whether there was probable cause.  In this case, the arresting officer "just knew, from experience" that the car was going 75 in a 70.  The opinion includes the officer's testimony, which really should be read.  The district court upheld the arrest, saying

He’s trained to estimate speeds.  His difficulty with measurements is
  immaterial to his estimate of speed as that did not depend on time or
  distance

However, the 11th Circuit Court (which overruled the district court) found contrarily that the officer was not "trained to estimate speed", he was allowed to guess, and trained to use radar (he did not use radar).  This does not mean that visual estimations are inadmissible, but they are vulnerable.  Thus there is a legal industry of fighting speeding tickets, and people who expertly know the scientific ground on which the evidence could be challenged.