Kinematics problem -- Cat chasing a mouse on a rotating disk

[Satvik Pandey]

Hello Chet



Isn't the velocity of cat always directed towards the mouse in the OP ?

What is OP?

 

[CAF123]

At a distance 'r' from center tangential velocity is rω or rV/R.

I don't understand the second equation. Yes, the tangential velocity of the cat is $v_{t, C} = r\omega_{C} = r \omega_{M} = ?$ Sub in what you got for the angular speed of the mouse earlier on. {t,C} stands for tangential velocity of the cat and M is a label for the mouse.

So, $\frac{dr}{dt}$=√(V^2-(rv/R)^2).

Right idea, but make the correction above in this formula.

After some algebraic calculation I found this-
∫dx/$\sqrt{R^{2}-r^{2}}$=$/\frac{V}{R}$∫dt. Upper and lower limits of LHS is R and 0 respectively. Upper and lower limits of RHS is T and 0 respectively.

You want the time it takes for the cat to reach R=28 having started at R=0. So you know the limits for R numerically. Your bounds for t are right. You just need to fix the integrand and then perform the integration.

 

[Satvik Pandey]

I don't understand the second equation. Yes, the tangential velocity of the cat is $v_{t, C} = r\omega_{C} = r \omega_{M} = ?$ Sub in what you got for the angular speed of the mouse earlier on. {t,C} stands for tangential velocity of the cat and M is a label for the mouse.

I just replaced ω.Here V is the velocity of mouse and R is 28cm(radius).I did this for calculating ω earlier in this discussion.
.

 

[Tanya Sharma]

What is OP?

Original poster/post .I was referring to post#1 i.e to the question.

 

[CAF123]

I just replaced ω.Here V is the velocity of mouse and R is 28cm(radius).I did this for calculating ω earlier in this discussion.
.

Ah okay, I misread the notation. So, V/R = 1/7. Sub this into your expression for the radial speed and the integral.

 

[Chestermiller]

Thanks Chet .

I am having difficulty in interpreting whether the pursuer’s velocity given in similar type of problems is radial velocity or instantaneous velocity .When the question says that the pursuer is always heading towards the target ,doesn’t it mean that the velocity given is instantaneous velocity ?

Here are two similar type of questions .


Q 1. A boy is on a boat, at a distance H from the shore, when he sees a girl (at the point on the shore where the distance is measured) running with a constant velocity u parallel to the shore. At that time, he moves towards her, with a speed v, in such a way, that the point of the boat is always pointed at the girl (so his vector is always pointing her way). Find the time of their meeting.

Q 2. Point A moves uniformly with velocity v so that the vector v is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity u < v. At the initial moment of time v is perpendicular to u and the points are separated by a distance l. How soon will the points converge?

What is your opinion regarding the velocity v given in the problems ? Do you think that the velocity v given in both the problems are the radial velocity or the instantaneous velocity ?

Maybe it would help if I started to set up the version of the problem that I'm thinking of. Let R and θ_M be the polar coordinates of the mouse at some particular time t, and let r and Θ_C be the polar coordinates of the cat at the same time t. The velocity of the cat is equal to V (a constant speed) times a unit vector in the direction from r,Θ_C to R,θ_M. Note, that in this version of the problem, Θ_C≠θ_M (except initially).

Please tell me if this makes any sense so far.

Chet

 

[Tanya Sharma]

Maybe it would help if I started to set up the version of the problem that I'm thinking of. Let R and θ_M be the polar coordinates of the mouse at some particular time t, and let r and Θ_C be the polar coordinates of the cat at the same time t. The velocity of the cat is equal to V (a constant speed) times a unit vector in the direction from r,Θ_C to R,θ_M. Note, that in this version of the problem, Θ_C≠θ_M (except initially).

Please tell me if this makes any sense so far.

Does the attached picture correctly represents the setup ? The brown vector represents the instantaneous velocity V of cat.Pink radial velocity V_r and green transverse velocity V_T .

 

[Tanya Sharma]

In the OP, v is the magnitude of velocity, that is, speed.

ehild

I would like to know whether in the two problems in post#32 'v' represents the magnitude of radial velocity OR magnitude of instantaneous velocity ?

 

[ehild]

I am having difficulty in interpreting whether the pursuer’s velocity given in similar type of problems is radial velocity or instantaneous velocity .When the question says that the pursuer is always heading towards the target ,doesn’t it mean that the velocity given is instantaneous velocity ?

Or it is speed given.

Here are two similar type of questions .


Q 1. A boy is on a boat, at a distance H from the shore, when he sees a girl (at the point on the shore where the distance is measured) running with a constant velocity u parallel to the shore. At that time, he moves towards her, with a speed v, in such a way, that the point of the boat is always pointed at the girl (so his vector is always pointing her way). Find the time of their meeting.

The speed of the boat is v, and the instantaneous velocity $\vec v$ points towards the girl.

Q 2. Point A moves uniformly with velocity v so that the vector v is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity u < v. At the initial moment of time v is perpendicular to u and the points are separated by a distance l. How soon will the points converge?

I do not understand the wording of this problem. What does it mean, moving rectilinearly? If the direction of the velocity changes with time, it is not "uniform". For velocities, you can not say that one is greater than the other unless they are parallel. You can compare the speeds.

What is your opinion regarding the velocity v given in the problems ? Do you think that the velocity v given in both the problems are the radial velocity or the instantaneous velocity ?

Velocity is a vector and independent of the coordinate system used. If it changes with time, you can speak about instantaneous velocity.
If you have a planar polar coordinate system, velocity has both radial and azimuthal (tangential) components, both depending on time. At the same time, the speed can be constant.
The radial velocity is the radial component of the velocity vector.

ehild

 

[Chestermiller]

Does the attached picture correctly represents the setup ?

Yes.

The brown vector represents the instantaneous velocity V of cat.Pink radial velocity V_r and green transverse velocity V_T .

I would have drawn it a little differently, with the angles measured from the x - axis, and the mouse running counterclockwise. But this is OK too.

Are you interested in working on this problem? The formulation of the problem should be interesting and instructive; the equations may not have an analytic solution, however.

Chet

 

[Tanya Sharma]

Are you interested in working on this problem? The formulation of the problem should be interesting and instructive; the equations may not have an analytic solution, however.

Sure

But first I would let Satvik finish his original problem .I have already derailed the thread :shy:.After he completes the problem I will get back to you .

Thanks Chet .

 

[Satvik Pandey]

Yes I solved it.
sin$^{-1}$$\frac{r}{R}$=V/R*T.
where the lower limit and upper limit of LHS are 0 and 28 respectively. On substituting the values I got T=11 seconds.THANK YOU ALL FOR GUIDING ME.

 

[Satvik Pandey]

Sure

But first I would let Satvik finish his original problem .I have already derailed the thread :shy:.After he completes the problem I will get back to you .

Thanks Chet .

The above discussion would be helpful for me even.After solving this I will try to go through the above discussion.Thank You.