A dog chasing the cat who is running along a striaght line at constant speed u . The dog moves with constant speed v, always heading towards the cat. Initially i.e. at t = 0 , the velocity of the dog and the cat are perpendicular and then initial perpendicular distance between them is 1. The dog catches the cat at ?

- mathslover

- schrodinger

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- mathslover

- AravindG

This is one of the classic questions on this topic .But I have to leav now .can definitely answer tomorrow is that OK @mathslover ?

- abb0t

If they are perpendicular, that meas they intresect at a 90ยบ angle.
the distance between them (i think hypotenusre) = 1

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- abb0t

How did I end up in physics section Lol.

- shubhamsrg

|dw:1359311516476:dw|
Let time taken be t,
then
(v cosx) t = 1 ...(1)
u = vsinx ..(2)
Just eliminate x to solve for t

- shubhamsrg

Oh wait, am missing on the fact that x is a variable.

- shubhamsrg

Getting too late here, I will get back to it later.

- UnkleRhaukus

is \(u>v,\quad u=v,\quad \text{or}\quad u

- shubhamsrg

u

- JamesJ

I am curious to see Aravind's solution. It's not hard to write down the equations of motion here. But solving them is not so obvious.

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- AravindG

@mathslover do you have final answer ? I wanted to check before replying

- JamesJ

Two major objections
1. The first one is the answer doesn't make sense. Suppose the velocities u and v are equal. Then it is clear the dog never reaches the cat. But according to the formula you have ended up with, the dog catches the cat in finite time of
\[ t = \frac{\sqrt{v^2 + u^2}}{v^2} = \frac{\sqrt{2}v}{v^2} = \frac{\sqrt{2}}{v} \]
2. The second one is you haven't explained your reasoning very well at all and there appear to be errors in the calculation.
For instance, why does the angle theta between their velocity vectors--or the y component of it--satisfy
cos theta = v/sqrt(v^2 + u^2)?
Or why does (v - u) cos theta = (v - u) . v / sqrt(v^2 + u^2) imply something (their relative velocity in the y direction?) equals v^2/sqrt(v^2 + u^2)?
And even if that were true, just because the relative displacement of two moving objects is <0,1>, by what kinematic equation is it the case that their relative velocity in the y direction can be directly deduced from that initial relative displacement, given that it is immediately clear that their relative displacement changes over time?
*****
So I am not satisfied we have the solution.

- abb0t

yoloswag

- JamesJ

But it is clear that the dog is NOT moving in a straight line. The dog is constantly changing direction.

- anonymous

thanks @jamesJ i got it the time period is \[\frac{ v }{ {v ^{2}-u ^{2}} }\]

- mathslover

I got it now , Thanks a lot every one...

- anonymous

and once again thanks to @mathslover for giving such a nice question keep it up!!

- mathslover

Thanks @harsh314 Just hope that I get a doubt soon in such a problem :)

- anonymous

may god bless you with such capabilites that you never get stuck in any question but for the sake of enrichment of our knowledge'thanks

- JamesJ

Wait. We still don't have a good solution to this problem. Harsh has written down a solution. But where's the proof?

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