finding the constant speed There were some traffic lights on the way from John’s house to his office. The distance between each traffic light was 450 metres. Each traffic light was function in a cycle as this, 35 seconds of green light followed by 5 seconds of yellow light followed by 35 seconds of red light. One morning, John rode his motorcycle to his office at a constant speed of 54 km/hour. He found that at each junction he met  green light. If he used the same way to go back from his office to his house, what is the maximum constant speed that he should ride in order for him to meet green light at all the junctions (except the first one)?
 A: I believe we have to assume there are a large number of lights, so that John has to hit each light at the same point of the cycle, or we cannot solve the problem.  
Hints:  What is John's speed, in m/sec?  Going to work, how long does John take to get from one light to the next?  We therefore assume that each light cycle is offset by that long from the one before.  How long is the light cycle?  On the return trip, what is the offset from one light to the next?  It may help to draw a picture showing the green and other color cycle of three or so lights.  What speed makes him take that long?
A: Drawing an x-t diagram was essential for me to figure this out.  You only need to draw the relationship between two adjacent lights to realize the answer.  Once you draw the path from the first green to the 2nd green, imagine the guy immediately turns around and goes back the way he came.  The speed at which he can do that and hit the next green of the light he came from is the answer to the question.
He will travel 450 m in 30 s at 54 km/hr.  This means that the green light cycles are offset by 30 seconds in the original direction.
Since the full light cycle is a total of 75s (35+5+35), traveling backwards he needs to span the 450m distance in 45s (75-30) to make the next green light going backwards.
So the answer is 10 m/s or 36 km/h.
A: We're going to make an x-t diagram to show you how this works.  Across the page is x (driving distance) and up the page is t, or elapsed time.  Use a scale of 5s per square vertically, and 45m per square across.  Draw the 35s green light cycle as a vertical line 7 squares high to the left of your page (he'll travel left to right initially).  Then find the point he will reach traveling 15 m/s for 450 m.  That will be 10 squares over (450m) and 6 squares up (30s = 450/15).  Draw a straight line between those two points (0,0), (450,30) to show the path of the motorcycle from left to right.  It's a straight line because it's a constant speed.  Since he started at the opening of the green cycle, this means he must reach the opening of the green cycle at the 2nd light (if he hit any other point in the light cycle, eventually he would miss a light).  Now consider what happens when he's traveling the other way.  He needs to get back to the first light when it turns green again.  When is it green again?  Draw the 5s yellow, then the 35s red cycles at x=0, stacked on the green cycle line (use different colors if you have them), then that's the green cycle start point again which he has to reach traveling backwards.  That's a total of 75s from t=0.  Draw a straight line back to the start of the next green cycle, from (450,30) to (0,75).  Now measure the slope of that line to get the speed he was traveling along it.  He's traversing 450m in 75-30 = 45s.  That's 10 m/s.
