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#1 2011-03-09 04:36:58

Reuel
Member
Registered: 2010-11-28
Posts: 178

Race cars - Differential Equation

I am still trying to work through some of these word problems but this one has me stuck.

"Leonardo had been in the lead ahead of his rival race car driver Michelangelo for some amount of time by a constant 3 miles. Just 2 miles from the finish, however, Leonardo's gas tank ran empty and his car decelerated ever after at a rate proportional to the square of his remaining speed. Within another mile, Leonardo's speed had halved exactly. If Michelangelo's speed remained constant, who won?"


Should this be set up as two different equations, one for each driver? The lacking of numbers for speeds is confusing. I dunno, that snowplow problem came after a little effort and this one seems as if it should be a lot alike, but I could use a little help with this one.

Thanks ahead of time. smile

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#2 2011-03-09 05:11:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

Hi Reuel;

I still do not have the required DE but a discrete analysis of it seems to yield that Michelangelo's car wins. Depending on the speed though.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2011-03-09 05:28:04

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

Hey bobbym, haven't talked to you in a few weeks. smile

Well, we know that the relevancy of the problem begins when Leonardo is 2 miles from the finish line and, at that instantaneous moment, when his car runs out of gas, Michelangelo is 3 miles behind him, so the total span is 5 miles.

When Leonardo's car begins to slow its deceleration is proportional to the square of his remaining speed. Is this a good way of stating it mathematically?

Negative because he is decelerating?

We also know that after one mile (which is half the distance) his speed had also halved. Does that mean his deceleration is actually linear?

Thanks. smile

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#4 2011-03-09 05:35:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

That is the DE I am playing with. I have solved it for the general case and am trying to determine the constants.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2011-03-09 05:43:29

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

I am very thankful for the help. I guess I am having an off-week.

More information: The difference in velocity between Leo and Mike before Leo ran out of gas was 0 which means that as Mike continues along, he does so at Leo's original velocity as Leo decelerates. That Leonardo reached half his speed in half the distance to the finish line, it sounds as though he might completely stop on the finish line and the whole time he is working up to doing so, Michelangelo has been closing the 3-mile gap between them at Leonardo's original speed.

Hmmmmmm...

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#6 2011-03-09 05:48:37

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

I am having trouble with determining the constants.

I am getting this but I do not understand the answer.


Solve these two simultaneously.

That is the point in which the two drivers finish at the same time.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2011-03-09 06:19:18

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

I don't understand the answer either. Or how you arrived at it.

I am working on it, too, but with none much luck thus far.

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#8 2011-03-09 06:20:19

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

Did you solve the DE?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#9 2011-03-09 06:24:10

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

I solved

to get

For time zero I simply set it equal to some number s so that C=s.

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#10 2011-03-09 06:25:20

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

Hi;

I think you forgot the square.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#11 2011-03-09 06:33:56

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

That could potentially pose as a problem. tongue


I see that.


Might all of this be best put into terms of position rather than velocity? Could that in some way be made to work with the positions of the cars?


In your's, where did

come from?

Last edited by Reuel (2011-03-09 06:37:42)

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#12 2011-03-09 06:42:20

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

At

v(0) = v where v is the initial velocity. Which we do not know.

v(1) = v / 2

Now solve for k and C


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#13 2011-03-09 07:45:18

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

Are you still working on this? Because I am completely stuck. I thought I had a good idea but... no.

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#14 2011-03-09 07:47:05

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

Hi;

Yes, I am. I am not sure of the solution at all. What are you stuck on?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2011-03-09 07:48:47

Reuel
Member
Registered: 2010-11-28
Posts: 178

Re: Race cars - Differential Equation

I was going to try to find position functions for both of them but Leonardo is hard to do because of limiting information. I keep ending up with too many unknowns.

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#16 2011-03-09 08:00:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Race cars - Differential Equation

I would expect that. What does yours look like?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2011-03-09 08:48:13

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: Race cars - Differential Equation

hi Reuel and bobbym,

I think the following might get you out of trouble:

Let distance from the 2 mile point be s (in miles)

so

when s = 0 , v = U (the common velocity until the moment the fuel runs out)

when s = 1 v = U/2

Now you have your constants.

M has five miles to go at U mph so you can specify t when he finishes.

So where is L then?

Write v = ds/dt and get an equation linking s and t to answer this.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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