I am working on a computer program that determines the minimum time it
will take for a subway train to go from one station to the next based
on a few parameters:

The parameters to the problem are all positive integers not greater than 1000.

    * d - the distance between stations, in metres
    * m - the maximum allowable speed of the train, in metres/sec
    * a - the maximum absolute acceleration of the train, in metres/sec2
    * j - the maximum absolute jerk, in metres/sec3 

The train must be completely stopped at each station and must move in
one direction at speeds not exceeding m. Acceleration can be positive
(forward) or negative (backwards) but its absolute value must not
exceed a. The last parameter, jerk, is the rate of change of
acceleration in either direction. That is, acceleration cannot
increase or decrease at greater than this rate. This parameter
prevents toppling the standing passengers.

For example, if the inputs are:

d = 1000
m = 70
a = 20
j = 1

The correct answer to 3 sig figs is 31.7

I have gotten close to this, but I am missing something. Basically, I
am looking for the math to figure out the problem.

Thanks in advance.

---

Request for Question Clarification by hedgie-ga on 27 Apr 2005 04:55 PDT

you say

 "jerk parameter
prevents toppling the standing passengers.."

But
 usually, acceleration, not jerk, measures toppling (by 2nd law of Newton)
 Max jerk may be describe constrains on the engine ..

So - what gives?

---

Clarification of Question by x911gt2-ga on 27 Apr 2005 09:12 PDT

Here is the URL to the problem...

http://acm.uva.es/p/v101/10124.html

---

Request for Question Clarification by hedgie-ga on 27 Apr 2005 21:32 PDT

Thanks for the URL. Nice picture of the train.

Now:

 Max jerk may be used to describe constraints on the engine.
 However, a , not j, measures toppling.

Please clarify the question.

---

Clarification of Question by x911gt2-ga on 27 Apr 2005 22:10 PDT

I can't give you any other clarification that what I have already
given. The exact problem I am working on is at the URL I have
provided.

As for jerk, from what I have read, it is in fact what causes a person
to 'topple' or experience whiplash....because jerk describes the rate
of change of acceleration. Meaning, if you accelerate to fast (rate of
change aka. jerk), then the person would 'topple' or experience
whiplash. This has nothing to do with the engine...atleast in this
case.

Check out this website for a definition of jerk and what it is used for.

http://math.ucr.edu/home/baez/physics/General/jerk.html

---

Clarification of Question by x911gt2-ga on 28 Apr 2005 12:16 PDT

philnj -- i appreciate all of your help.

I have the model, but I am running into the same issues you are --
arriving at a solution. I've contacted a few of my professors, but
they were unable to help me as well.

What you were saying about increasing the acceleration as quickly as
possible. Unfortunately, you can only increase the acceleration by the
jerk rate, which for their example was 1m/s. I actually read that
*most* trains stay in the 2/ms jerk rate range as a general principle.

hedgie -- first off, i found out what jerk is...its you. 

I came to this place looking for help, not a lecture or for someone
who would try and insult my intelligence. I am not a physics major, I
am a programmer, so the physics part of this is all new to me. I don't
need to switch schools, but I do recommend you take a course on how to
interact with people in an acceptable manner.

You were right about one thing. This is a homework assignment.
However, my professor was clear in stating that we can use any
availble resources to solve our problem. I am not asking you or anyone
else to write the program. I just wanted help in figuring out the
physics of the problem. The assignment for me is programming, not
physics.

Anyways, you did succeed in leaving a negative impression of what
google researchers are supposed to be. Maybe they should do a better
job of screening. As for your peace offerings, I don't need them.

I'm sorry, I can't help you with this interesting problem, but I have one comment:

That's some subway train you've got there!  70 m/s top speed = about
160,000 miles an hour!
I hope you're using maglev and running in an evacuated tunnel.
(And don't expect me to shovel the coal!)

A guess that makes j=2 now if you count me.

:-)

Actually, 70 m/s = ~156mph...

Easy mistake to make :-) 

Google calculator works wonders...

://www.google.com/search?q=70+m%2Fs+to+mph

Thanks though!

D'oh!
Make that j=3.

I hope you have success.

I agree with hedgie, in which case you can use the equations of
motion.  Where did 31.7 seconds come from?  It seems too long, unless
it's a round trip.

Ian G.

This is an interesting diversion from what I am being paid to do!

I don't have a total solution, but I think I know what the issues are.
First, it is a third order polynomial in time because of the change in
the acceleration.  I don't yet know if you are required to solve this
polynomial for time.
Second, the parameters are limits, so you have to check that you don't
exceed your limits in any step of the calculation.  For example, it
you accelerate at your limit, your final "Cruising speed" may be
greater than your limit.
Third, there are some assumptions you must make to solve for the
initial conditions.  One, you have to assume that the change in
acceleration is the same in every instance.  Second, you have to
assume that the time of acceleration and deceleration is the same.

I started with a simple graph of acceleration.  It starts at zero,
ramps up (at a rate j) to a steady value not to exceed a.  It then
ramps down at the same rate to a value of zero.

I also split the problem in two halves.  If you calculate the time it
takes to get half way, you can forget about the second half of the
acceleration graph that is the same as the first part but has negative
values.  Is this correct?

I broke the graph into pieces timewise.  By taking the integral of the
acceleration curve, you get the velocity curve.  Take the integral of
the velocity curve and you have the distance curve.  You know the
distance, so all you have to do is solve for time.

Easier said than done?

Maybe I'll think about this overnight.

OK. I slept on it.

The polynomial algebra gets a little scary, but the problem boils down
to a couple concepts.  First, there are four variables.
1.  The rate at which the acceleration changes (j' limited by the
input parameter j).
2.  The length of time spent increasing your acceleration (time interval t0 to t1)
3.  The length of time spent at a constant acceleration (time interval t1 to t2).
4.  The time at which you reach constant cruising speed (time interval
t2 to t4).  BTW, you assume that you lower your acceleration at the
same rate that you raise you acceleration (t3=t2+t1).  You also divide
the problem in half.  If you know how long it takes to get half way
there, you double it to get to your destination.

If you think about it, these are the choices a train driver would make
if his hand were on the throttle.

Second concept:  You have four variables and a set of given limits. 
And you are asked to find the MINIMUM time to get between two
stations.  That means maximizing acceleration without overshooting
your destination.

If you slog through the math, you can integrate the acceleration
equations for the velocity and integrate velocity to get position. 
Position, velocity and acceleration are all subject to limits.

That is as far as I've gotten so far.  Is any of this helping, or am I
talking to myself?  Regardless, I'm enjoying this challenge.

Your concepts are correct. However, my problem is I am stuck with the math.

Be more specific.  Are you having trouble modeling the problem or
solving for the answer?  I've got the model, but am having trouble
working toward the solution.  If you need what I can do, I'll be glad
to cover it in more detail.

I'm currently working on how to deal with the limits.  I'm thinking
that it is best to increase your acceleration as rapidly as possible
and reach your speed limit if you can.  But at each step, you have to
check that you have not exceeded your limits.  There are three cases.

1.  You spend all your time increasing your acceleration and
decreasing your acceleration.  There is not enough distance or your
maximum velocity is too low to get to your acceleration limit.

2.  You can reach your acceleration limit, but there is not enough
distance to level off at a cruising speed.

3.  You can get to your max velocity and must stay at that speed until
you have reach the half way point.

I wish I could draw you about a half-dozen pictures.  But I have no
place to post them.

Wow, usually researchers are a model of decorum.  But looks like this
one is pissed at something.  It is a physics problem, dude.  No reason
to criticize someone like that.

The explanation of the maximum principle looks like what we are after.
 And the value we are calling jerk is simply the slope of the
acceleration curve.  It integrates to a third order time term.  What
it is or what causes it has no bearing on the solution.  I think it
was thrown in to make the problem more interesting (difficult).  It
worked.

philnj -- i appreciate all of your help.

I have the model, but I am running into the same issues you are --
arriving at a solution. I've contacted a few of my professors, but
they were unable to help me as well.

What you were saying about increasing the acceleration as quickly as
possible. Unfortunately, you can only increase the acceleration by the
jerk rate, which for their example was 1m/s. I actually read that
*most* trains stay in the 2/ms jerk rate range as a general principle.

hedgie -- first off, i found out what jerk is...its you. 

I came to this place looking for help, not a lecture or for someone
who would try and insult my intelligence. I am not a physics major, I
am a programmer, so the physics part of this is all new to me. I don't
need to switch schools, but I do recommend you take a course on how to
interact with people in an acceptable manner.

You were right about one thing. This is a homework assignment.
However, my professor was clear in stating that we can use any
availble resources to solve our problem. I am not asking you or anyone
else to write the program. I just wanted help in figuring out the
physics of the problem. The assignment for me is programming, not
physics.

Anyways, you did succeed in leaving a negative impression of what
google researchers are supposed to be. Maybe they should do a better
job of screening. As for your peace offerings, I don't need them.

I am sorry of I insulted your intelligence,
when I just to provoke it to some activity.
If you would care to use it, you could actually see that Portjagin 
principle provides an algorithm:

a/j  is time to reach max a
a/v       "           max v
d/v       "   coasting 'remaining' distance

then you repeat that backwards,
add the times
and you are home, I eman at the next station

but of course - you do not need that

so sorry again. You do not have to change schools,
 It just meant that  that you do not
need to all accept statement posted on the web
 without some critical evaluation ..

I also apologize for some missing words in my msg:

"..It just meant that  that you do not
need to all ..."

 should have been:
".. It just meant that  that you do not
need to ACCEPT all .."

(may be I should wake up first, next time, before responding).

 and 

 perhaps you can try to  look at it from the other side:

What should a researcher do, 
when s/he tells customer
that his assumption is incorrect,
and customer just keeps repeating it,
without any intelligent argument to support his assertion?
 An URL to some web-page is not an argument, since web is full of nonsense
 particularly when it comes to physics.

Are you seriously still ranting about this?

You state that you repeadetly told me that my assumption was wrong.
However, this was not an assumption. Jerk, which is the rate at which
acceleration changes (im sure you already knew that), is what causes
toppling, whiplash and so forth. That is why when engineers build
rollercoasters and subways, they take jerk into account.

Just because you are a 'researcher' doesn't mean you are always right.
I even asked my physics professor, who used to work as a roller
coaster engineer to give me an explanation. Here it is:

"Jerk is important when evaluating the destructive effect of motion on
a mechanism or the discomfort caused to passengers in a vehicle.  The
movement of delicate instruments needs to be kept within specified
limits of jerk as well as acceleration to avoid damage.  When
designing a train the engineers will typically be required to keep the
jerk less than 2 metres per second cubed for passenger comfort." --
Phil Gibbs

Also, check out wikipedia, which gives a good explanation:

http://en.wikipedia.org/wiki/Jerk

"Jerk is used at times in engineering, especially when building roller
coasters. Some precision or fragile objects?such as passengers, who
need time to sense stress changes and adjust their muscle tension, or
suffer e.g. whiplash..."

It's pretty obvious that jerk does come into play in this problem.
Yes, acceleration can make someone fall if it becomes too great...but
on a smaller scale, the rate at which it changes (JERK) can have the
same effect. In the case of this problem, this is the situation.

Is that intelligent enough for you?

I am not claiming to be an expert on physics, but I did do my research
on this problem. I went as far as I could and that is why I came here
for assistance.
The first URL I gave you was to the problem. That way you could see
what I was talking about. I wasn't saying you were wrong. But now I
am.

So am I going to look at it from your side? No. Why? Primarily,
because I am offering YOU, the researcher, MONEY to help with my
problem. When it comes down to it, the customer is always right (in a
sense). You seriously should think about how you approach people from
now on. I noticed you deleted your rude posts -- good move to cover
yourself. I hope you mend your ways and become a bit more open minded.
Good luck.

x911: Are you seriously still ranting about this?
 H:     No. Not seriously.

x911: I am offering YOU, the researcher, MONEY

H: When researcher switches from RFC (on top) to comments (on the bottom)
   s/he does not work for money. Standards are more relaxed, since comments
   are free. The reason may be that s/he cannot answer the question (which was 
   not the case here) or because asker does seem  cooperative or polite etc...

X911: jerk does come into play
H:      Indeed - it allows people to adjust, grab something or tension 
        up their muscles. It just does not topple people, it just allows the
        toppling force to surprize them.

X911:  I noticed you deleted your rude posts

H:      I deleted nothing.
        If you re-read the dialog, you will notice who was
        calling people names. (In spite of my handle, 
        I am actually a human being, not a hedgehog).

conclusion:
If you use the sketched up algorithm -
 you need to check the factors,like
is it a/j or 2a/j or 4a/j or so,
 and really understand the solution. 
Sketch  is not intended to be a full solution. 
This, limited answer is free. If your consciousness will move you so,
you may same some money to a charity, victims of civil wars - perhaps.

H.

You were the one who flipped out on me. Remember? And yes, you deleted
the comments that you made in the first place that started this whole
debacle. If you didn't, then why is your comment that WAS between
phil's comments at 8:18 and 9:50 missing? Anyways, like I said before,
I don't want your answers or your input. So, please do us both a favor
and move on. You already made your impression and you can't change it.
You're lucky I don't report you to google. I would, except for the
fact that your exceptionally rude post is suddenly *missing*.