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#1 2012-11-26 01:40:34

pokrate
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Physics olympiad question

A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed u. Without any delay, the dog starts with running with constant speed v>u to catch the cat. Initially, v is perpendicular to u and L is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of v, u and L.

#2 2012-11-26 02:02:36

pokrate
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Re: Physics olympiad question

Please see my approach so far. I have used very small interval dt when cat has moved a distance of udt and dog vdt.

I am stuck with strange equation.

Thanks in advance


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Last edited by pokrate (2012-11-26 02:23:48)

#3 2012-11-26 03:27:38

bobbym
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Re: Physics olympiad question

Hi;

Your problem is a pursuit curve.

http://mathworld.wolfram.com/PursuitCurve.html

http://curvebank.calstatela.edu/pursuit2/pursuit2.htm

google for more about pursuit curves. See if that helps you.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#4 2012-11-26 04:10:50

pokrate
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Re: Physics olympiad question

Thanks for the links for enlightening me. Request you to take a look into this problem and lets together try to come up with some solution.

#5 2012-11-26 04:19:41

bobbym
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Re: Physics olympiad question

Hi;

Right now I am knee deep in several other problems as well as administrative matters. I already solved one of these before here, take a look at it

http://www.mathisfunforum.com/viewtopic … 10#p144710


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#6 2012-11-26 05:08:08

pokrate
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Re: Physics olympiad question

Thanks again, but I will be extremely happy and will work on my own, if you or someone can just take a look at my approach to comment if it right or wrong ?

#7 2012-11-26 18:48:00

scientia
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Re: Physics olympiad question


Suppose at time t the dog is running at angle θ to the cat's direction of motion. The component of the cat's velocity parallel to that of the dog is ucosθ so the velocity of the dog relative to that of the cat is



where s the separation between the animals at time t. (Note that the RHS is positive as
.) Suppose the dog catches the cat after time T. Integrating gives



Now, the component of the dog's velocity parallel to that of the cat is vcosθ, so when the dog catches the cat we have





Substituting the integral into the previous equation gives



#8 2012-11-26 20:22:28

pokrate
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Re: Physics olympiad question

scientia wrote:


Suppose at time t the dog is running at angle θ to the cat's direction of motion. The component of the cat's velocity parallel to that of the dog is ucosθ so the velocity of the dog relative to that of the cat is



where s the separation between the animals at time t. (Note that the RHS is positive as
.) Suppose the dog catches the cat after time T. Integrating gives



Now, the component of the dog's velocity parallel to that of the cat is vcosθ, so when the dog catches the cat we have





Substituting the integral into the previous equation gives



This solution is copied from physicsforum website, and is non-calculus based, as integration is cancelled out.

#9 2012-11-27 01:19:38

pokrate
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Re: Physics olympiad question

If your approach is right, then why this approach is not right as shown in the image :


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