Determine time after which 4 persons meet in given scenario

[vcsharp2003]

Homework Statement

Four persons K, L, M, N are initially at the four corners of a square of side d. Each person now moves with a uniform speed v in such a way that K always moves directly towards L, L directly towards M, M directly towards N and N directly towards K. The four persons will meet after how much time?

Relevant Equations

v= d/t i.e. velocity = displacement/time
Relative velocity of A wrt B = velocity of A - velocity of B

The point of confusion is deciding the direction each persons sets out in i.e. velocity direction of each person. Knowing this will probably help in getting the solution.

At t=0, I can say that velocity of each person is as shown in diagram below.

 

[Vanadium 50]

I don't see an attempt, so I will give you one word - which is one more than I should. Symmetry.

 

[vcsharp2003]

I don't see an attempt, so I will give you one word - which is one more than I should. Symmetry.

It seems they would meet at the center of the square, but then direction of velocity of each person would be changing every instant. In such a dynamically changing situation, how would I calculate the time? The direction of $\vec v$ would be changing in direction all the time. If each person traveled a curved path to the center, then determining total distance would be difficult. If I knew that distance then I could use the simple equation of velocity = distance/time.

 

[PeroK]

It seems they would meet at the center of the square, but then direction of velocity of each person would be changing every instant. In such a dynamically changing situation, how would I calculate the time? The direction of $\vec v$ would be changing in direction all the time. If each person traveled a curved path to the center, then determining total distance would be difficult. If I knew that distance then I could use the simple equation of velocity = distance/time.

What do you think happens to the distance between K and L if K moves towards L and L moves at right-angles to this direction?

 

[Vanadium 50]

Let your answer be T(side length) and the length of the square be a, that is your answer is T(a).
Why would I write it this way?

 

[vcsharp2003]

What do you think happens to the distance between K and L if K moves towards L and L moves at right-angles to this direction?

Distance between K and L should decrease with time. Initial distance is d between them and it finally reduces to 0.

 

[PeroK]

Distance between K and L should decrease with time. Initial distance is d between them and it finally reduces to 0.

We can all see that. What is the initial reduction in distance (initial relative speed)? Can you calculate that using trigonometry?

 

[Vanadium 50]

Can you calculate

I think at this point you have plenty of hints. You should try and set up a calculation.

 

[Vanadium 50]

I meant calculate the answer, not some random thing.

 

[vcsharp2003]

We can all see that. What is the initial reduction in distance (initial relative speed)? Can you calculate that using trigonometry?

Below is velocity of K relative to L.

 

[PeroK]

Below is velocity of K relative to L.

That's not what I asked you to calculate. I asked you to calculate the rate at which the distance between K and L is decreasing.

The magnitude of the relative velocity is something else.

 

[vcsharp2003]

That's not what I asked you to calculate. I asked you to calculate the rate at which the distance between K and L is decreasing.

The magnitude of the relative velocity is something else.

I think the initial distance between them is d which gradually becomes zero and also point K would be approaching point L at velocity v always, so time taken to meet should be d/v.

Is that right?

 

[PeroK]

I think the initial distance between them is d which gradually becomes zero and also point K would be approaching point L at velocity v always, so time taken to meet should be d/v.

Is that right?

We all know that. Can you really not calculate the rate at which the distance initially reduces between K and L?

 

[vcsharp2003]

We all know that. Can you really not calculate the rate at which the distance initially reduces between K and L?

If we take K and L, then velocity of K is always going to be perpendicular to velocity of L, no matter what path they take. This means the velocity of approach between K and L will always be v since K has no component along the perpendicular. So my answer to your question is v, which is the velocity of approach between K and L.

 

[PeroK]

If we take K and L, then velocity of K is always going to be perpendicular to velocity of L, no matter what path they take. This means the velocity of approach between K and L will always be v since K has no component along the perpendicular. So my answer to your question is v, which is the velocity of approach between K and L.

If you can justify that, then you've got the answer.