How do I find the time taken by dogs to meet each other?

[Thiru07]

Homework Statement

Sam has three dogs sitting at the vertices of an equilateral triangle. The length of each side of the triangle equals to s meters. Sam gives the command "Start!" and each dog starts to run with constant speed v meters per second. At each moment, each dog is running towards the dog just right to him (in counter-clockwise direction). Therefore, their trajectories are forming some spirals that converging to one point as shown in the attachment.
How long does it takes dogs to meet each other after the command "Start!"?

Homework Equations

Time = Distance / Speed

The Attempt at a Solution

For Speed = 5 and Side = 2
Approximate answer is 0.266667. But I don't know how to get that.I'm struggling to find the distance the traveled by the dogs.
Once I have got the distance , I can plug it in the above formula to get the time.

I know I'm certainly missing something .

Any help would be appreciated.

 

[haruspex]

Pick two dogs. What is their relative speed?

 

[Thiru07]

Pick two dogs. What is their relative speed?

As the speed is same for all the dogs , relative speed would be zero right?

 

[haruspex]

As the speed is same for all the dogs , relative speed would be zero right?

No, I mean the magnitude of the relative velocity.

 

[Thiru07]

No, I mean the magnitude of the relative velocity.

No, I mean the magnitude of the relative velocity.

Let's say speed of each dog is 5 meter per second.
Now I'm picking Dog at vertex a and Dog at vertex b.
Is it 5+5 = 10 ?

 

[Thiru07]

Let's say speed of each dog is 5 meter per second.
Now I'm picking Dog at vertex a and Dog at vertex b.
Is it 5+5 = 10 ?

I'm sorry for not knowing basic concepts.
Now I'm trying to r

Let's say speed of each dog is 5 meter per second.
Now I'm picking Dog at vertex a and Dog at vertex b.
Is it 5+5 = 10 ?

I'm sorry about not knowing basic concepts.
Which concepts should I learn/know to solve this problem?

Thanks

 

[Simon Bridge]

relative velocity: http://hyperphysics.phy-astr.gsu.edu/hbase/relmot.html
remember that velocity is a vector.

the lynchpin for most people is to realize what shape the dogs (bugs whatever) make at later times.

 

[haruspex]

Let's say speed of each dog is 5 meter per second.
Now I'm picking Dog at vertex a and Dog at vertex b.
Is it 5+5 = 10 ?

Consider the initial position. Do you understand components of vectors? What is the component of the first dog's velocity in the direction from that dog to the second dog? What is the second dog's velocity along the same line?

 

[ehild]

The dogs are always at the corners of an equilateral triangle, each of them running towards the next.

What is the velocity of the second dog (red) with respect to the first one(green)? The arrows represent the velocities. You can take that the green arrow points to the positive x direction.

 

[Thiru07]

vector

Consider the initial position. Do you understand components of vectors? What is the component of the first dog's velocity in the direction from that dog to the second dog? What is the second dog's velocity along the same line?

Yes . Every vector has two components x and y. To find the magnitude of a vector we take square root of sum of the square of change in x component and that in y component.
X is the component of the first dog's velocity in the direction from that dog to the second dog. And second dog's velocity along the same line is same as x component of the first dog but the direction is opposite.

 

[Thiru07]

The dogs are always at the corners of an equilateral triangle, each of them running towards the next.

View attachment 107437
What is the velocity of the second dog (red) with respect to the first one(green)? The arrows represent the velocities. You can take that the green arrow points to the positive x direction.

Is it the result of division of velocity of dog 1 by cos 60 i.e twice the velocity of dog 1 ?

 

[ehild]

Is it the result of division of velocity of dog 1 by cos 60 i.e twice the velocity of dog 1 ?

No.

 

[Thiru07]

No.

I got that result using 90-60-30 triangle property . What am I doing wrong? can you please help?

 

[haruspex]

I got that result using 90-60-30 triangle property . What am I doing wrong? can you please help?

In ehild's diagram, the component of dog 1's velocity along the green arrow towards dog 2 is obviously its entire speed v, right? Dog 2's velocity along the red arrow has a component in the green direction. What is that component?

 

[Thiru07]

In ehild's diagram, the component of dog 1's velocity along the green arrow towards dog 2 is obviously its entire speed v, right? Dog 2's velocity along the red arrow has a component in the green direction. What is that component?

That's x and it's direction is opposite to Dog 1's velocity . is magnitude of Dog 2's x component and Dog 1 's velocity same here?

 

[haruspex]

is magnitude of Dog 2's x component and Dog 1 's velocity same here?

Certainly not. The two speeds are the same, but the directions are different. The component of a velocity in some other direction must be less in magnitude than that of the whole velocity.
Let's label the corners of the triangle A, B, C corresponding to dogs 1, 2, 3.
Drop a perpendicular from the red arrow tip (point D) to the line AB connecting dogs 1 and 2 (the line in the direction of the green arrow), meeting it at point E. If the red arrow has length 2, what is the distance from the base of the red arrow (point B) to point E?

 

[Thiru07]

Certainly not. The two speeds are the same, but the directions are different. The component of a velocity in some other direction must be less in magnitude than that of the whole velocity.
Let's label the corners of the triangle A, B, C corresponding to dogs 1, 2, 3.
Drop a perpendicular from the red arrow tip (point D) to the line AB connecting dogs 1 and 2 (the line in the direction of the green arrow), meeting it at point E. If the red arrow has length 2, what is the distance from the base of the red arrow (point B) to point E?

Ok. Distance from the base of the red arrow (point B) to point E is 1.

 

[haruspex]

Ok. Distance from the base of the red arrow (point B) to point E is 1.

Right. So if the red arrow is a velocity vector of magnitude v, what is its component in the BA direction.

 

[Thiru07]

Right. So if the red arrow is a velocity vector of magnitude v, what is its component in the BA direction.

In BA direction it's component is x and in ED direction it's component is y right?

 

[haruspex]

In BA direction it's component is x and in ED direction it's component is y right?

What you call the components is irrelevant. What is the magnitude of the component? The answer is some fraction of v. Look at how you answered in post #17. Apply the same logic.

 

[Thiru07]

What you call the components is irrelevant. What is the magnitude of the component? The answer is some fraction of v. Look at how you answered in post #17. Apply the same logic.

Magnitude of that component is 1. Assuming that this magnitude I have got is right then the relative speed of dog 1 and dog 2 will be the difference of magnitude of velocity of dog 1 and the magnitude I got in post # 17?

 

[haruspex]

Magnitude of that component is 1.

No, as I posted, the answer is some particular fraction of v. It cannot be the whole of v, but something less.
If the red arrow represents the velocity vector of the second dog, its length BD is the magnitude of that vector, v. To find the component of that in the direction towards dog 1, we drop a perpendicular from the tip of that arrow to the line AB, meeting it at E. The component is the vector BE. If BD has length v, how long is BE?

 

[Thiru07]

No, as I posted, the answer is some particular fraction of v. It cannot be the whole of v, but something less.
If the red arrow represents the velocity vector of the second dog, its length BD is the magnitude of that vector, v. To find the component of that in the direction towards dog 1, we drop a perpendicular from the tip of that arrow to the line AB, meeting it at E. The component is the vector BE. If BD has length v, how long is BE?

Why is it not v*cos60 ?
What am I missing? Do I have to use some formula here to find the length of BE ?

 

[haruspex]

Why is it not v*cos60 ?

It is indeed that. Or, more simply, v/2. If you said so earlier I must have blinked and missed it.
So, knowing that dog 1 has speed v towards dog 2 along that line, and dog 2 has a speed v/2 towards dog 1, what is their closing speed?

 

[Thiru07]

It is indeed that. Or, more simply, v/2. If you said so earlier I must have blinked and missed it.
So, knowing that dog 1 has speed v towards dog 2 along that line, and dog 2 has a speed v/2 towards dog 1, what is their closing speed?

That must be v/2 .I hope it's correct.

 

[haruspex]

That must be v/2 .I hope it's correct.

No. If I walk towards you at speed v and you walk towards me at speed v/2, how rapidly are we approaching each other?

 

[Thiru07]

No. If I walk towards you at speed v and you walk towards me at speed v/2, how rapidly are we approaching each other?

3v/2 which is the relative speed you had asked me in the beginning right?

 

[haruspex]

3v/2 which is the relative speed you had asked me in the beginning right?

Yes.
Now, is it clear to you from ehild's post that until they meet the dogs will always form an equilateral triangle; it just shrinks and rotates?

 

[Thiru07]

Yes.
Now, is it clear to you from ehild's post that until they meet the dogs will always form an equilateral triangle; it just shrinks and rotates?

Finally, I got what relative speed is .Thanks a lot. Now every two dogs' relative speed is 3v/2?
Now, I have got the relative speed ,my next step is to find the distance between those two dogs and then find the time?

 

[haruspex]

Finally, I got what relative speed is .Thanks a lot. Now every two dogs' relative speed is 3v/2?
Now, I have got the relative speed ,my next step is to find the distance between those two dogs?

You know the original distance, and the relative speed along the line joining two tells you how rapidly that distance diminishes.

 

[Thiru07]

You know the original distance, and the relative speed along the line joining two tells you how rapidly that distance diminishes.

Wow. Got it. I can't thank you enough for your step by step guidance.

 

[ehild]

Because of symmetry, the dogs are at the corners of a equilateral triangle all times. The triangle rotates and shrinks with time, and it becomes a point at the end, at the centre of the original triangle. Both the time when the dogs meet and their trajectory can be easily obtained in a polar system of coordinate. The sides of the original triangle are of length a.

The radial and tangential components of the velocity vector of a dog (represented by the red dot now) are v_r=dr/dt=-vcos(30°)= -v√3/2. The tangential component is v_θ=rdθ/dt=vsin(30°)=v/2.
The radial component is constant, so the radius is r=r₀-√3/2 vt. At t=0, r=r₀=a/√3. When the dogs meet, r=0, the time is
t=2/3 a/v.
The equation for θ is dθ/dt=0.5v/r Substituting r=a/√3-√3/2 vt and integrating, we get the time dependence of the angle. The trajectory is obtained by eliminating the time, but also using that dr/dt=dr/dθ dθ/dt, that is,
$-v\frac{\sqrt{3}}{2}=\frac{dr}{d\theta}\frac{v}{2r}$
Integrating, the trajectory is $r(\theta)=\frac{a}{\sqrt{3}}e^{-\sqrt{3} \theta}$