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**pokrate****Member**- Registered: 2012-11-16
- Posts: 13

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.

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**pokrate****Member**- Registered: 2012-11-16
- Posts: 13

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

*Last edited by pokrate (2012-11-25 03:23:48)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,776

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**pokrate****Member**- Registered: 2012-11-16
- Posts: 13

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.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,776

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

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**pokrate****Member**- Registered: 2012-11-16
- Posts: 13

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 ?

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

Suppose at time

where *s* the separation between the animals at time *t*. (Note that the RHS is positive as

Now, the component of the dog's velocity parallel to that of the cat is *v*cos*θ*, so when the dog catches the cat we have

Substituting the integral into the previous equation gives

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**pokrate****Member**- Registered: 2012-11-16
- Posts: 13

scientia wrote:

Suppose at timetthe dog is running at angleθto the cat's direction of motion. The component of the cat's velocity parallel to that of the dog isucosθso the velocity of the dog relative to that of the cat iswhere

.) Suppose the dog catches the cat after timesthe separation between the animals at timet. (Note that the RHS is positive asT. Integrating givesNow, the component of the dog's velocity parallel to that of the cat is

vcosθ, so when the dog catches the cat we haveSubstituting the integral into the previous equation gives

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

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**pokrate****Member**- Registered: 2012-11-16
- Posts: 13

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

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