# Homework Help: How do I find the time taken by dogs to meet each other?

Tags:
1. Oct 14, 2016

### Thiru07

1. The problem statement, all variables and given/known data
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!"?

2. Relevant equations
Time = Distance / Speed

3. 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 travelled 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.

#### Attached Files:

• ###### WP_20161013_13_50_54_Pro.jpg
File size:
16.4 KB
Views:
88
Last edited: Oct 14, 2016
2. Oct 14, 2016

### haruspex

Pick two dogs. What is their relative speed?

3. Oct 14, 2016

### Thiru07

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

4. Oct 14, 2016

### haruspex

No, I mean the magnitude of the relative velocity.

5. Oct 14, 2016

### 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 ?

6. Oct 14, 2016

### Thiru07

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

Thanks

7. Oct 14, 2016

### Simon Bridge

8. Oct 14, 2016

### haruspex

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?

9. Oct 14, 2016

### 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.

10. Oct 14, 2016

### Thiru07

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.

Last edited: Oct 14, 2016
11. Oct 14, 2016

### Thiru07

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

12. Oct 14, 2016

### ehild

No.

13. Oct 14, 2016

### Thiru07

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

14. Oct 14, 2016

### haruspex

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?

15. Oct 14, 2016

### Thiru07

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?

16. Oct 14, 2016

### haruspex

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?

17. Oct 15, 2016

### Thiru07

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

18. Oct 15, 2016

### haruspex

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

19. Oct 15, 2016

### Thiru07

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

20. Oct 15, 2016

### haruspex

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.

21. Oct 16, 2016

### Thiru07

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?

22. Oct 16, 2016

### haruspex

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?

23. Oct 16, 2016

### Thiru07

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

24. Oct 16, 2016

### haruspex

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?

25. Oct 16, 2016

### Thiru07

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