Two Trains, a Bird and Crash (Sept. 2021 Challenge Question)

It takes 94.59 hours for the trains to meet....flying at 25 mph the bird would have covered 2,364.86 miles...

The train from NY would have covered 1,891.8 miles and be somewhere in the middle of Colorado....mountain time... 3.93 days later

The NY train left 3 hours before the other train....recalculating...12:00 noon

So the ny train and bird leaves 3 hours before the sanfran train...so they have a 60 and 75 mile respective head start...the trains combined speed is 20+15=35mph for 3440 miles = 98.28 hours to meet + the three hours already transpired...for a total of 101.28 hours...the bird will have travelled 2532.14 miles....4.22 days will have passed...that would be 4 days 5 hours and 28 min....sept 5 and 3:00pm + 5 hours and 28 min = 8.28 pm - 3hrs for timezone = 5:28 mountain time on sept 5....

6:28 pm

September is the equinox when day and night is equal length...and the Sun is directly over the Equator...

The bird will have travelled 2532.14 miles, and the collision would happen sept 5 at 6:28 pm....

This is it! Leave it to an eagle to understand the mathematics of a bird.

Well done.

No time zones in 1870.

A Central Pacific steam locomotive leaves San Francisco at 15 miles per hour at 3 p.m., September 1, heading east.

A Union Pacific steam locomotive leaves New York City at 20 miles per hour at 3 p.m., September 1, heading west.

The length of track between the two stations is approximately 3,500 miles. The trains do not make any stops along the way.

At 3 p.m., September 1, a bird leaves New York City, following the tracks, at 25 miles per hour. It flies until it reaches the eastbound train, when it instantly reverses direction.

When it reaches the westbound train, the bird reverses course again, and flies back to the eastbound train. The cycle repeats until the bird is killed in a fiery, catastrophic train wreck.

How many miles did the bird fly? When do the trains collide?

NY train starts 3 hours earlier (EST).

The 1st 3 hours, NY train travels 60 miles, distance between trains at 6 PM EST = 3440 miles.

At 6 PM EST trains approach at 15+20 = 40 mph, time to meet = 3440/40 = 86 hours = 3 days, 14 hours.

Trains meet at 8AM EST Sept 5. Bird flies 86 hr x 25 mi/hr = 2150 miles

Since the bird and the first train start at the same time in NY, the flight time is 86+3 hrs.

Hence the bird flies 2225 miles.

Right, thanks!

No time zones in 1870.

That doesn't mean that the time was the same everywhere, they just hadn't named it yet....they did have clocks everywhere...

My mention of no time zones at post #8 was also marked as off topic by someone.

I just wish they'd explain why they think it's off topic.

The difference between us is that I believe that the simplest interpretation of the phrase "leaves San Francisco at 15 miles per hour at 3 p.m." is that the train left 3 hours after solar noon in SF.

You seem to me to believe that it should be interpreted as meaning that the train left 3 hours after solar noon in NY (Or that the trains both left at the same time relative to some other arbitrary longitude.).

The reason I believe that the OP may believe that the lack of time zones is relevant is that the time zones were introduced because of the problems caused to train time tables by their absence.

That was my mistake. The point about time zones is correct.

However, I think it somewhat violates the spirit of the challenge, which should be a fairly clean mathematical exercise. I think the OP needs to clarify any assumptions about time difference.

Solar Eagle may already have done a revised calc based on 3.23 hours versus 3.

(My estimate of Kearney, NE, was based on highway mileage; railroad mileage would differ somewhat.)

15+20=40?

Oops, I still had 25 mph in my mind from that dang bird!

Let it be assumed that the trains are Big Boy and the Allegheny so that the crash will be spectacular.

It is requested that the answer be specific, but the question excludes information necessary for that specificity. That is: 1) How long does it take for the bird to reverse direction? 2) How long is the bird?

It is unrealistic to assume that the bird reverses direction instantaneously (or that it accelerates/decelerates instantaneously). In the time it takes to reverse direction the trains will have moved closer together. This will become particularly critical in the last moments before the crash when the bird is vibrating back and forth between the trains. In those moments the turning time of the bird will exceed its flight time.

It is also unrealistic to assume that the bird is a dimensionless point. A bird has length. If the bird is, say, 6 inches long, then the bird's flight will end when the trains are 6 inches apart. Or maybe not! If the bird is in the middle of a turn as the trains come together, then we also need to know the thickness of the bird. In that case, the bird's flight will end when the trains are the thickness of the bird apart.

The bird is a holographic projection...

The United States railroads didn't introduce time zones 'til 1883. Until then everyone set their clocks by the sun. So I think you need to look at the precice longitude of NY and SF.

Calculating the exact time of the crash you would need to know the route. I'm guessing that it winds more in th west.

So probably meet in Wyoming....

https://www.arcgis.com/apps/Cascade/index.html?appid=e679f80b19ed4482b10910f2a918946e

Maybe around Cheyenne or Laramie..

Joining of the railroads May 10, 1869 at Promontory point Utah ...

..."In 1870 it took approximately seven days and cost as little as $65 for a ticket on the transcontinental line from New York to San Francisco; $136 for first class in a Pullman sleeping car; $110 for second class; and $65 for a space on a third- or “emigrant”-class bench."...

https://www.gilderlehrman.org/sites/default/files/inline-pdfs/Transcontinental%20Railroad%20Fact%20Sheet.pdf

https://www.smithsonianmag.com/arts-culture/traveling-style-and-comfort-pullman-sleeping-car-180949300/

In the absence of time zones SF is about 3.23 hours behind NY.

So the NY train will have travelled 64.55 miles when the SF train sets off.

And the trains would then need to travel 3435.45 towards each other at a relative speed of 35 mph

That would take 98.16 hours, and, the flight time of the bird would be 101.38 hours.

The bird would have travelled 2534.58 miles.

The trains crashed at 8: 23 pm New York time 4 days after the start of the journeys.

Without detailed knowledge of the route it is not possible to determine local (to the crash) time.

The OP states:"Length of track...".

No ambiguity there.

...but it is still 'approximately'...

Huh?

My answer is based on the track being 3500 miles long.

The problem with identifying the exact local time is that we can't determine the longitude of the crash if we don't know how much more "wiggly" the track is in the West than the East.

And, of course the local time depends solely on the longitude in the absence of time zones.

They never would meet and the bird must still be flying. As the trains don't make stops they will stop by themselves, as a steam engine must carry its fuel and Wikipedia says range was 75-150 miles, depending on load (how many cars and passengers, which we don't know either).

Imaginary trains don't need no fuel...

Just a little soul...

A little off topic - but what the hell

At collision time t, being

Dsf = 15 t , Dnw = 20 t, Dsf + Dnw = 3500

t = 100 hours

If the bird is a point and reversing direction instantaneous it flies 25 x 100 = 1000 miles

..." 25 x 100 = 1000 miles"...??

I know how much is 25 x 100. Sorry. And you know what I meant!

Excellent

The bird flies 2500 miles. The trains collide 100 hours after departure time. Sept 4th 3.10pm

100 ÷ 24 = 4days.......4 + 1 = sept 5

Agreed I'm a day short.

Recalculated and here is the answer. The trains collided on Sept 5th at 3.39pm. The poor bird flew 2500 miles to get squished.

No time zones or twisty tracks were considered as they were not mentioned in the posting.

I calculated the trains traveled 1500 miles and 2000 miles to the collision point. Anyone concur on the above?

Yes.See my post#24.

A very simple calculation unless it is made more difficult by speculation and hair splitting.

Since the actual distance is some 1000 mi shorter, you would need to subtract 500 miles from each to calculate the actual time at the collision location...1500 miles west from NY which would be Maxwell, Nebraska...around an hour and a half time difference...give or take

The OP clearly stated:

"The length of track between the two stations is approximately 3,500 miles. The trains do not make any stops along the way."

"TRACK MILES = 3500".

Pretty clear to me.

The trains did not leave the track,except maybe when they collided.

As far as I know,trains don't fly,at least not yet.

Yes it's called an airliner....

Where are the tracks?

Ain't no train without tracks.

Just a modular plane.

The approaching speed of the trains toward each other is is 35 mph.

It will take 100 hours for the trains to meet.

The NY train will travel 2000 miles in 100 hours,the San Fransisco train will travel 1500 miles in 100 hours.

The bird will have been flying for 100 hours at 25 mph so he will have flown 2500 miles,assuming instantaneous reversal.

The trains will meet 2000 miles from NY,and 1500 miles from San Fransisco.

This is presuming no stops for water tanks,(usually every six miles),and a clock independent of time zones and no acceleration time.

Off Topic?How so?

All previous answers are incorrect in one or more respects.

Because of the time difference, the NY train has a 3-hr head start. During that time, the NY train travels 60 miles and the bird 75 miles.

Once both trains are moving, 3440 miles remain to be covered at a combined approach speed of 35 mph. This takes 98-2/7 hours (3440/35).

NY train: (98-2/7 x 20) + 60 = 2025-5/7 miles.
SF train: (98-2/7 x 15) = 1474-2/7 miles.
Bird: (98-2/7 x 25} + 75 = 2532-1/7 miles.

Around Kearney, Nebraska?

Not so.

Your answer is wrong.

I stipulated "a clock independent of time zones."

There were no time zones at the time.

I stand by my correct and simple analysis.

You are not the owner of the thread, so you cannot make stipulations. Clocks were set by the sun, so there was 3.36 hours difference (25000/3500 = 7.14; 24/7.14 = 3.36).

Without further stipulation by the OP,I understood the starting time to be

instantaneous,(C), not zone or location dependent.

Further constraints and more information need to be provided by the OP,to clear the

matter up.

He asked for the answers to be specific,so he should be more specific with the

conditions of the question.

The way the question is stated,it leaves the door open for multiple correct

interpretations.

Fun question! Simple at first sight, but in second thought I see that there are still many aspects to consider like the institution of time zones :-)

My thoughts are wandering.

The trains and bird are obviously imaginary, as SolarEagle pointed out. They don't need to refuel, and the bird can fly for at least ~100hrs straight. HOWEVER, they do crash at the end! So they are not fully imaginary :-)

No imagine that it is not fully hypothethical, and that we let go of some other imaginary aspect: impuls and momentum :-)

A single impact of a tiny bird (assume pidgeon, 500g) versus a train does not affect the train very much. However, as the distance decreases, the number of impacts per hour increases dramatically!

Above you can see the paths of the trains and the bird (in blue): it bounces back and forth, and iterates into decreasing small steps in time.

At a distance of 1 foot the collision frequency is around 50 per second. At a separation of one inch, the frequency has increased to more than 100 per second.

Now 100 'bullets' of half a kg per second will impact the train; creating a spectacular view JUST before the crash between the trains :-) A high-speed camera (I know, hypothetical in 1870) will show this.

....I have come in the tail end and see many answers relate to the precision applied to speed and size of the bird, and it's physical ability to change direction.....

...but to my mind the correct answer will depend on the "exact" distance between stations - rather than "approximately 3,500 miles" ...

The distance between the stations doesn't matter, it's the length of track traveled...and the bird is following the track...

The distance from NY to San Francisco is 2572 miles...as the crow fly's...

By car it's 2902.2 miles...

Trains from New York City to San Francisco, CA cover the 2560 miles (4129 km) long trip taking on average 76 h 25 min....

My_atlas_agrees_with_2902_highway_miles.__(Something_is_wrong_with_this_editing_window:_it_doesn't_recognize_spaces_or_carriage_returns.__I'll_try_to_submit_this_and_see_what_happens.)o_itonm

It_also-doesn't_recognize_backspaces_and_deletions.__I_wonder_where_the_error_resides.

(IIf-the_crow-flies_distance_is_2572_miles,_how_can_the_railroad_distance_be_only_2560_miles?__Maybe_the_Elon_Musk_Straight-line_boring_distance_shaves_off_the_extra_12_miles?"a

Must be altitude to fly direct creates a larger arc than a direct route that goes through mountain tunnels....

Apparently the train stations were 12 miles closer together in 1870 than the airports are now.

At 3 p.m., September 1, a bird leaves New York City, following the tracks, at 25 miles per hour. It flies until it reaches the eastbound train, when it instantly reverses direction.
When it reaches the westbound train, the bird reverses course again, and flies back to the eastbound train. The cycle repeats until the bird is killed in a fiery, catastrophic train wreck.
How many miles did the bird fly? When do the trains collide? (Be specific!)

OK specific ... the bird, which must in fact become a resonance, assuming the laws of motion are in abeyance, would suggest that the bird would become the closest thing man has known to infinity - both positive and minus, as vectors, ... a picosecond, 1 x 10−12 before collision = X dimension? And so on.
Bloody stupid question meant for 12 year olds.

We weren't taught sums of infinite series, limits, and calculus when we were twelve years old. Is this the New Math curriculum? I do remember that the basic train/bird problem (without the time zones) was a bonus question on a Grade 11 or 12 math test.

Who said the bird was killed?

The first point of contact between the two trains would be the cow catcher,which

protrudes in front of the train very near the track.

A bird capable of instantaneous direction reversal could surely be quick enough to avoid

being crushed by the mass above him.

" The cycle repeats until the bird is killed in a fiery, catastrophic train wreck."

Touch'e.

Good catch.

I didn't see that part of the op.

It must have been lost in the bifocal zone.

https://www.youtube.com/watch?v=x1_xKMOMnDU&list=RDCMUCTRxnMudmOGNw0g4gZf5D8g&start_radio=1&rv=x1_xKMOMnDU&t=25

It is a hypothetical question. Nothing should be assumed or added. We have the speeds, distance, and time. There are no other complications of time zones mentioned or twisty tracks.

Town A (San F) is 3500 miles from town B (NY). Why worry about time zones and track constraint where none exist?

And 1870 could be 7.10pm and the train driver could be 5' 14'' tall. And there are easier ways to kill a bird than using two trains. Bit of a over kill if you ask me, (which you did not, so don't). The bird started of as a 2.2lbs bird and ended up after all that flying back and forth as a dead weight shredded 1kg bird.

Ah but the bird size and weight were not stipulated and more importantly do we assume that the pointy bit of the beak is the point of measurement?

It certainly is a hypothetical question. But here and now it is not a question in a math exam, where you would know you'd have to calculate a few numbers and you're done. Instead, it is a question on an engineering website, an environment of different minds with various backgrounds and experiences. I think mentioning these specific cities is done on purpose, and there is room for interpretation and out-of-the-box thinking.

The specificity of the mentioned cities gives a connection to the real world, with real-world complications like time differences and time zones :-)

After seeing the first few answers on such a challenge question, I more enjoy the discussion and alternative solutions :-)

One point forgotten of course is the fact the in the minutes, seconds and micro seconds before collision the poor bloody bird must vibrate, resonate, echo, whatever you wish to call it, at an ever increasing frequency until an infinity minus - minus infinity before the finite crash.
MY niece was over, and she be a Prof of Math, and she laughed and said that the bird would actually fly close to an infinite number of kilometres, and explained why..until she said Uncle Mickey do you want another bottle of wine...at which point I said "Yep I knew that" ... she then pointed out that the question was meant for kids, not grumpy old engineers.
Poor kid was never smart enough to be an engineer.

How is it that "Prof of Math" does not understand a finite converging series?

She does, quite thoroughly, but I had tuned out and was concentrating on a perfectly good 2008 Malbec.
To bloody smart for her own good. If she watched the All Blacks flog the Wallabies she will give you a blow by blow math analysis of the bloody game ... great kid but to dumb to be an engineer.
BTW I tuned out during that as well.

However brilliant she is, it seems she must have slept through the class on convergent series. Theoretically, you are correct that the bird flies back and forth an infinite number of times, but the sum of those infinite number of decreasing distances is not infinity. The sum tends to a limit.

Yep, it shows we are not all the same and we don't see things the same. But I did manage to identify the bird. The bird was a rare dumbarsed lesser spotted nuthatch ostrich. Length, beak to tail, 540mm, wingspan, 500mm, weight, 700g. Regions found; South West Asia minor, central and southern Africa. West America. Last sighting was on a train track between San Fransisco and New York in 1870. It is now believed to be extinct.

I say the question is a misprint because the stations are approximately 2500 miles apart, then and now...not the 3500 miles stated.....so I declare this question to be out-of-order! This thread is therefore out-of-order!

Another aspect of bird flight so far unmentioned unless in the posts I didn't read fully: airspeed vs ground speed. The bird may "follow" the track, but at what altitude and in what wind?

The bird may stop for food, too, just as the train needs fuel (my post a few days ago). I know some birds fly thousands of miles over ocean, but they can land on the ocean or islands, and I don't know if/what they eat on the journey. Bird people may know the calories needed per mile, considering bird weight which changes with time as food consumed for energy to fly, air resistance, wind and airspeed.

This question is not only about math and birds but about problems: the value of a solution, the need for precision, the possibility of multiple or no solutions, relative confidence in each, and the ability to see a problem as needing more info. Also, personality and position of the person(s) who want the solution, and the time scale available to get one... and more.