Interesting post. Just had me pondering it all day and I thought I’d propose another way to go about this: visually. Open THIS in another tab. What you have is a plot of distance vs. time. The black line is shown with a constant slope of 20mph.
I have plotted two points on that line, (t1,d1) and (t2,d2). Imagine these as snapshots of your journey. You travel along the black line and at some moment check your clock and odometer. These “checkpoints” would be like t1/t2 and d1/d2. You compare your travel time to your distance traveled and determine that d/t = 20mph.
At each of these snapshots, you decide to maximize your speed over the same distance traveled so far. If you were traveling along on the black line, this means that you take a sharp left and travel infinitely fast along the dotted line until your distance is twice what you’d traveled so far. Since you traveled infinitely fast, you spent no time and your new coordinates are either (t1, 2d1) or (t2, 2d2).
Note that both of these points lie on a new line, shown in green. This line’s slope indicates the final rate achieved. We can now see that:
r = (2d1)/t1 = (2d2)/t2
But d1/t1 and d2/t2 are equal to d/t = 20mph because they lie on the original constant sloped line. So we know that the green line’s slope is 2 d/t = 2 20mph = 40mph.
Just thought it might be interesting to see this portrayed another way. I’ve shown what happens if you travel infinitely fast and use a vertical line to head on up to the final distance. If you traveled less than infinitely fast, your line would lie between the original rate (20mph) and our shown theoretical maximum. This area is the solution set and is highlighted in light green.
Very nice! This illustrates the idea that at any time during the journey, the average speed up to that point constrains the possible average speeds for the whole journey.
I thought I’d point out that merely the first point (t1, d1) suffices to construct the second (green) line, since the lines both start at the same point (the origin).
Thanks! And yes, I could have left it at one point on each line since the origin counts, but thought two points might help drive the point (no pun intended!) home, lest one point appear to be a “lone solution”—two helps show that the green line is actually a maximum to a whole solution set rather than just a line created from one data point.
Interesting post. Just had me pondering it all day and I thought I’d propose another way to go about this: visually. Open THIS in another tab. What you have is a plot of distance vs. time. The black line is shown with a constant slope of 20mph.
I have plotted two points on that line, (t1,d1) and (t2,d2). Imagine these as snapshots of your journey. You travel along the black line and at some moment check your clock and odometer. These “checkpoints” would be like t1/t2 and d1/d2. You compare your travel time to your distance traveled and determine that d/t = 20mph.
At each of these snapshots, you decide to maximize your speed over the same distance traveled so far. If you were traveling along on the black line, this means that you take a sharp left and travel infinitely fast along the dotted line until your distance is twice what you’d traveled so far. Since you traveled infinitely fast, you spent no time and your new coordinates are either (t1, 2d1) or (t2, 2d2).
Note that both of these points lie on a new line, shown in green. This line’s slope indicates the final rate achieved. We can now see that:
r = (2d1)/t1 = (2d2)/t2
But d1/t1 and d2/t2 are equal to d/t = 20mph because they lie on the original constant sloped line. So we know that the green line’s slope is 2 d/t = 2 20mph = 40mph.
Just thought it might be interesting to see this portrayed another way. I’ve shown what happens if you travel infinitely fast and use a vertical line to head on up to the final distance. If you traveled less than infinitely fast, your line would lie between the original rate (20mph) and our shown theoretical maximum. This area is the solution set and is highlighted in light green.
Very nice! This illustrates the idea that at any time during the journey, the average speed up to that point constrains the possible average speeds for the whole journey.
I thought I’d point out that merely the first point (t1, d1) suffices to construct the second (green) line, since the lines both start at the same point (the origin).
Thanks! And yes, I could have left it at one point on each line since the origin counts, but thought two points might help drive the point (no pun intended!) home, lest one point appear to be a “lone solution”—two helps show that the green line is actually a maximum to a whole solution set rather than just a line created from one data point.