Forums >Running 101>Math Nerds - distance calc, track with gates

The track I have access to has 4 gates that intend to block the inside 4 lanes. The gates are about 10m inside the start and end of each straight.

I read here: http://www.maa.org/mathland/mathtrek_08_11_03.html that each lane out from lane 1 adds 6.70 meters of extra distance per lane per lap.

I did "400m" intervals last night - started at start of straight in lane 5, ran the straight, cut to lane 1 gradually as passed second gate, swung back out to lane 5 on the back straight, repeat back to start.

There really isn't a significant difference in the effect on the workout, just kind of curious if anyone a bit more clever at math than I am could figure the actual distance out.

Come all you no-hopers, you jokers and rogues

We're on the road to nowhere, let's find out where it goes

Simple application of the Pythagorean Theorum, a^2 + b^2 = c^2.

"a" is the length of the gate from the curb to where you have to run around the end.

"b" is the distance from the gate to the point down-track at which you arrive back at the curb after running around it.

"c" is the hypotenuese of the triangle, the actual angled path across the track you run from the outer end of the gate back to the track curb in lane 1.

Without the gates there, each time you pass the spot you would run 2Xb. But because of the gates you must run 2Xc. So to know how long your "400m" intervals are, measure a and b, calculate c. Subtract b from c to get the additional distance you have to run, remembering that you must double this for each gate (you run "c" going out around the gate, and again coming back).

So if there are 4 such gates on the track, the distance is about 400m + 8(c-b).

Couldn't you just do the workout in lane 5 using the staggered start?

Runners run.

modified: 6/20/2012 at 7:27 AM

Couldn't you just do the workout in lane 5 using the staggered start?

That probably makes sense - now I just have to figure out how to do that - I don't know much about tracks.

MTA: if I did do that, I assume that it would make my 200m jog recoveries different? I dunno. I have to read up on how tracks work. I don't get out there much.

Spaniel - thanks for the figures.

We're on the road to nowhere, let's find out where it goes

I thought that the Pythagorean Theor*e*m dealt with right triangles, not curves.

That probably makes sense - now I just have to figure out how to do that - I don't know much about tracks.

MTA: if I did do that, I assume that it would make my 200m jog recoveries different?

There should be staggered markings in each lane just past the common start finish line for the 400m "two turn" start. If you start at the 400m start in lane 5 (or any lane) and finish at the common finish line, that's 400m. If you really wanted to keep the jog to 200m then after the finish line you could jog the turn, turn around and jog back to the 400m start in lane 5.

Runners run.

I thought that the Pythagorean Theor

em dealt with right triangles, not curves.

Yes, but spaniel used the Pythagorean *Theorum*.

Either I don't understand the configuration -- entirely possible -- or Spaniel's answer makes no sense to me. To get an exact number we'd have to know what "cut to lane 1 gradually as passed second gate" means in terms of the curve traversed. And the answer wouldn't be as simple as an application of the Pythagorean Theor*e*m. It sounds to me like a very close approximation would involve the difference between arc length on a half circle and arc length on a half ellipse (not a simple function, also close and not exact because the gates are 10m inside the start and end of the straights).

A diagram would help in either case.

Or you could do what Mikey says.

Jeff

modified: 6/20/2012 at 8:54 AM

The track is 400 meters when run .3meters (approx 1 foot from the curb)

The straightaway is 84.39 meters (and that wouldn't change)

The radius of the corner is 36.5 meters (when run at the curb).

Each lane is supposed to be 1.22 meters wide

If, the gate is 4 lanes wide (it would be 4.88 meters or 16 feet) out from the curb, the radius (if running then entire run in the 5th lane) would be about 41.4 meters.

So, the math for running the entire run in the 5th lane (.3 meters to the right of the lane 5 lane marker) would be:

straight away + left radius + right radius

(2 times 84.39) + (1/2 of (2 * pi * 41.4) * 2)

= 168.78 meters + 130meters + 130meters

=429meters

By doing the shortcut as you described, there would be a reduction in the meters, and I can't figure that out while still doing some type of work here at the office

Cheers,

Brian

2014 Goals:

#1: Do what I can do. **<DOING>**

#2: 365 Hours training **<NOPE, INJURED>**

There should be staggered markings in each lane just past the common start finish line for the 400m "two turn" start. If you start at the 400m start in lane 5 (or any lane) and finish at the common finish line, that's 400m. If you really wanted to keep the jog to 200m then after the finish line you could jog the turn, turn around and jog back to the 400m start in lane 5.

Cool. Thanks, mikey.

We're on the road to nowhere, let's find out where it goes

modified: 6/20/2012 at 8:56 AM

'Course my method only works for 400s or 200s. You want to start doing 1000s or 600s or miles something for which there is no staggered start, you're back to doing some mid-level Euclidean Geometry--in other words way over my head. And who wants that when they're running?

At that point I'd probably either take my workout to the road and do it by time, or just do it in lane 5 but do it by time only and estimate the total distance once I'm done.

(Or I'd move the gates and put them back when I'm done.)

Runners run.

modified: 6/20/2012 at 9:06 AM

Simple application of the Pythagorean Theorum, a^2 + b^2 = c^2.

"a" is the length of the gate from the curb to where you have to run around the end.

"b" is the distance from the gate to the point down-track at which you arrive back at the curb after running around it.

"c" is the hypotenuese of the triangle, the actual angled path across the track you run from the outer end of the gate back to the track curb in lane 1.

Without the gates there, each time you pass the spot you would run 2Xb. But because of the gates you must run 2Xc. So to know how long your "400m" intervals are, measure a and b, calculate c. Subtract b from c to get the additional distance you have to run, remembering that you must double this for each gate (you run "c" going out around the gate, and again coming back).

So if there are 4 such gates on the track, the distance is about 400m + 8(c-b).

Spaniel's description of cutting the corner (in a hyptoneuse style straight line has merit, but I don't quite see the math. There would be 4 hypontenuse's run (gate to apex and apex to gate for left radius, and gate to apex and apex to gate for right radius).

a = radius of corner (41.4meters), center to gate

b = radius of corner (36.5 meters), center to apex

41.4^ + 36.5^ = c^

1714 + 1332 = c^

3046 = c^

c= 55.19 meters

so, by using my previous logic, the math would be

(straightaway x 2) + 2 left hypotenuse + 2 right hypotenuse

= (84.39 x 2) + (55.19x2) + (55.19x2)

=168.78 + 110.38 + 110.38

= 389.54meters...

ugh!!!

The problem is that you can't run a straight line from the gate corner to the apex because the curb is in the way (or I totally screwed up my math thinking). Likely the latter.

2014 Goals:

#1: Do what I can do. **<DOING>**

#2: 365 Hours training **<NOPE, INJURED>**

I use this and run a lap and then figure out how long it was after. But to do that you have to stay in the same lane all the way. Which I figure is more polite anyway as I didnt pay for the track surface that they are trying to preserve by blocking off the first 4 lanes.

**I have become Death, the destroyer of electronic gadgets**

"When I got too tired to run anymore I just pretended I wasnt tired and kept running anyway" - dd, age 7

My local track has signs up saying "please don't train in lanes 1 & 2". I guess this is because they get rather more wear than the other lanes.

If you stick in one lane then you can get a pretty good estimate of the length of a complete lap in that lane by just pacing out the length of the 400m stagger in that lane.

I was in the '10 Items or Less' line at a grocery store in Cambridge, MA, holding a basket with - obviously - less than 10 items.

The young man in front of me had a basket piled high with much more than this limit. So the girl at the register takes a long look at him, said, "Say... you must attend either Harvard or MIT."

"Wow," said the guy. "How did you know that?"

Without missing a beat, the girl replied, "Because this is a '10 Items or Less' line. And you either go to MIT, and can't read, or go to Harvard, and can't count."

Forums >Running 101>Math Nerds - distance calc, track with gates