PennsyFan said:
I'm sorry to say it, Alan, but your math is wrong. It takes 3 minutes longer to reach track speed, but the train is moving in the time spent speeding up. It's not as if the train is spending three minutes sitting, and then suddenly speeds up at the normal rate. Also, Batallion 51 said it took 3 minutes total to reach track speed, rather than 3 minutes more.
The math would look a lot more complicated than anything anyone has suggested. You'd have to find the rate of acceleration and calculate out how fast the train is going at each phase of acceleration. If anyone here feels like taking a shot at it, go for it.
Whatever the answer turns out to be, however, it is still clear that the second engine wouldn't make that much of a difference. It would certainly be nowhere near an hour.
You asked for it!
Given: The train with one P42 takes 3 minutes to get to 80mph. The train with two P42’s takes 2 minutes to get to 80 mph.
Find: The time added to the trip for each 0 to 80 mph acceleration by the one unit train.
Assumption: Acceleration is constant from 0 to 80 mph.
OK, here we go!
Lets take a train with one P42 accelerating from rest to 80mph in 3 minutes. 80mph is 117 feet per second. Moving from rest to 117fps in 180sec. is an acceleration of 0.65 feet per second per second (fps2). Good so far? OK!
Now, the distance covered by an accelerating vehicle is equal to the acceleration (A) times one half the elapsed time squared or:
S=0.5 x A x time x time (take my word for it)
Plugging in our numbers, we get:
S=0.5 x 0.65 x 180 x 180 = 10,560 feet. <== that is EXACTLY 2 miles! Amazing!
So, the train with one P42 takes 10,560 feet to go from rest to 80 mph. It covers that distance in 3 minutes.
Now, lets say a train with two P42’s can accelerate to track speed, 80mph, in 2 minutes (one minute faster than the one unit train). That’s an acceleration of 0.98 fps2. The distance covered to get to track speed is:
S=0.5 x 0.98 x 120 x 120 = 7,040 feet.
So, the train with two P42’s is at 80 mph after just 7,040 feet. To cover the same distance the train with one P42 used to get to track speed, this train will move an additional 3520 feet (10,560 – 7,040) at 80 mph (117 fps). The time needed to move 3,520 feet at 117 fps is:
3,520/117 = 30 sec
So, to cover 10,560 feet starting from rest, the train with two P42’s takes 120 seconds for acceleration to track speed in 7,040 feet, and another 30 seconds at track speed to cover the remaining 3.520 feet, or a total of 2.5 minutes.
Ah ha! The train with one P42 takes three minutes (180 seconds) and 10,560 feet to get to 80 mph. The train with two P42’s covers that same 10,560 feet in 2.5 minutes. Therefore, dropping one P42 adds one half minute to the running time for every start from rest to 80 mph.
Isn’t physics fun!