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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.
#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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#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.