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

[Chestermiller]

OK.

Does the picture correctly represents the situation in OP ? Blue dot represents the cat and red represents the mouse .Green is radial and pink is transverse velocity .

Does the brown vector correctly represents the instantaneous velocity in the OP ?



But if the instantaneous velocity vector of the cat was towards the mouse ,that means the velocity has only the radial component and in that case how is it possible for the cat to catch the mouse ?

Only if the angular velocity of the cat is the same as that of the mouse.

Doesn't the cat need to have an angular(transverse) component of velocity as well ?

Yes. The cat must have an angular component of velocity in both cases. In the OP, its angular velocity component is specified as being the same as that of the mouse. In the case where the cat's total resultant velocity is always directed toward the mouse, the cat's angular position falls behind that of the mouse.

Chet

 

[Tanya Sharma]

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 ?

 

[ehild]

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

ehild

 

[ehild]

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.

Here v is speed. In the OP, v is also the magnitude of velocity, that is, speed. They are constant, but the direction of the instantaneous velocity changes.

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 ?

Here the velocity also changes direction, so it is not 'uniform'. Uniform can be the speed.

ehild

 

[Satvik Pandey]

Okay, but that is a peculiar notation since θ is usually associated with the angular direction. Sub in the expression for velocity in tangential direction and reexpress the radial velocity in terms of the time derivative of the radial displacement (that is displacement of cat from the centre of the circle).

At a distance 'r' from center tangential velocity is rω or rV/R.
For a small distance dr tangential velocity can be considered as constant.
So, \frac{dr}{dt}=√(V^2-(rv/R)^2).
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.

 

[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.