3 cars spaced equal distance thus forming equalateral triange with side equal to a...?

Question

starts moving with equal speed v
1 moving towards 2
2towards3
3 towards 1
what is the time they all meet at a point


question was asked in entranse exam.

Answer 1

I would look at the relative speed. 1 moves toward 2. 2 is moving somewhat towards 1 as it moves toward 3. Use trigonometry to calculate the component of 2's motion that is toward 1 (I think it's v/2). Then add that to 1's speed (v). Then divide the side length (a) by that total speed (3v/2).

So I say, t = 2a / 3v

Answer 2

This is an exercise in relative motion. They will meet if and only if their respective intercept angles remain fixed as the triangle shrinks. Since they started in the three corners of an equilateral triangle, the three cars need to maintain that 60 degrees angle relative to each other at each point of the shrinking triangle. In which case, the three cars will collide in the center of the triangle.

Thus, the vector of one car needs to be V = vx + vy; where vx = V cos(30) and vy = V sin(30) since the vector V bisects a corner angle. The time the three of them will meet in the center is found from t = S/V; where S = sqrt(x^2 + y^2) and x = 1/2 L, and L is the edge length of the triangle when the cars first start out. y = S sin(30); so we solve for S in S^2 = (1/2 L)^2 + [S sin(30)]^2. S is the distance from one point of the triangle to the center where the three cars will meet.

The keys to working this problem are to recognize the internal 60 degree angels of the triangle establish the relative angle of intercept that must be kept constant. And to recognize that, once you've solved that resulting velocity vector V for one car, that V is the same in magnitude for the other two; so time to intercept will just be t = S/V because all three will hit that point (and each other) in the same time.

Answer 3

This is an interesting question.

Basicaly the travel path of the three cars will look like a swirl pattern within the triangle.

As car 1 travels towards car 2, its direction will constantly be changing since car 2's direction is changing as well.

Answer 4

if they continue to follow each other, they dont meet because they are going the same speed.

ive seen this question asked like this

if you have 3 ants in each corner inside of a triangle and they all move towards the center of the triangle, will they ever run into each other?