Distance, time, and speed studied

Distance, time, and speed are interrelated. Which travel route is best to take from point A to point B shown on the chart below? Miles and kilometers are interchangeable in this question and have no effect on the answers.

The grey route goes straight for 10 miles (km) from point A to point B. The blue route goes one mile (km) on the grey route and then cuts off to the village at the top and returns to the grey route one mile (km) before point B. The resulting distance when using the blue route is 11 miles (km) total, with 9 miles (km) in the center instead of 8 miles (km) when using just the grey route. The blue route goes past the village with no reduction in the 65 mi/hr (km/hr) speed limit.

Here is a chart summarizing the routes, with the resulting average speed shown when going the speed limit for the total distance using both the grey route and the blue route.

       

Assuming that the speed limits posted on the routes are never exceeded, which route is best to take involves answering the following questions:   

1.     Which travel route(s) get(s) you there

a.     fastest?

b.     earliest?

2.     When using both route(s), the following factors make the trip from A to B have the same time of arrival, and they are

a.     The minimum speed must you travel on the 9 mile (km) blue route portion?

b.     The maximum distance on the blue route, assuming that you maintain the 65 mi/hr (km/hr) speed limit on it?

The first question uses “fastest” and “earliest” as meaning different things, and they are. Your driving speed determines which is “fastest,” and the time spent in travel determines which is “earliest.” They are used interchangeably in normal conversation, even though not so.

Here is a chart that shows you the answer:

 

Question 1, a. is answered by saying that going the speed limit on both routes results in the highest average speed, as shown previously above, thus it is the fastest rate of travel is possible. Question 1, b. is answered by saying that traveling both routes at the speed limit also gets you there the soonest, saving you 26 seconds. 

Question 2, a. is answered in the light blue stripe in the table above showing a minimum speed of 61.87 mi/hr (km/hr) on the blue route section gets you there at the same time as going straight through on the grey route. Question 2, b. requires the additional table of values below to show the answer.

That table shows the maximum distance in miles (km) that use of the speed limits on both routes can get you to point B at the same time as if you went straight from A to B on the grey route. It is 11.455 miles (km), and subtracting the 2 miles (km) that you must travel on the grey route to get to and from the blue route, the blue route is 9.455 miles (km), which has an addition of .455 miles (km). If only .5 miles (km) is added to the 9 mile (km) blue route, you will arrive 2 seconds later than if you stayed on the grey route. If a mile (km) is added to the blue route, you will lose 30 seconds using it.

This short essay serves as a preface to a much longer one, still in development, that adds temperature to the equation and gets into relativity.