# Thread: Circular Track (First Meeting and Second Meeting)

1. ## Circular Track (First Meeting and Second Meeting)

Natsu and Gray both run around a circular track. They both started to run at the same time. Gray started at the westernmost point of the track and runs counterclockwise, and Gray takes about ninety-five seconds to run one lap of the track. Natsu runs clockwise and he passes Gray for the 1st time after eleven seconds. Natsu runs at 3 meters per sec (speed) and takes 80 seconds to run 1 lap of the track.

What are Natsu's coords when he passes Gray for the 1st time?
What I did:
I knew that it took Gray about 95 seconds to run 1 full lap (which is equal to 2pi radians). So 2pi/95 rad/sec is his angular speed.
While Natsu's angular speed is then 2pi/80 rad/sec
Therefore speed = angular speed * radius
I knew one of the speed was 3, so 3 = (2pi/80)*r, and solved for r to be 240/2pi = 120/pi
Then I knew θ = angular speed * time, so I found the θ to be 11 * 2pi/95

There I put everything into this: rcos(θ+pi), rsin(θ+pi)
Where I got the coords: (-28.52, -25.40) for when Natsu meets Gray for the first time. (So I got the answers correct here, the second question is where I need help).

What are Natsu's coords when he passes Gray for the second time?
This is the part I got lost on:
Ok, so I know Natsu's angular speed is 2pi/80
While Gray's angular speed is 2pi/95

What I tried doing was to find the time, through θ = angular speed * time
So then θ = 2pi/80 * t (for Natsu)
And 2pi - θ = 2pi/95 * t (for Gray)

Though I intended to solve for t (time) and plug in t+11 to the coordinate equation: rcos(θ+pi), rsin(θ+pi)
Though I got lost on how to solve for the angle this time... which then gave me troubles finding time.

I knew that the angle had to be something like 2pi - θ at least... which just basically means the angle they swept through to meet the second time.
Anyway, what should I do from here? Unless there's another easier way.

Answers for the second part: (34.256, 16.898)

Thanks

2. ## Re: Circular Track (First Meeting and Second Meeting)

Hey Chaim.

Your total track length is 2*pi*r = 240 metres. If you can convert your co-ordinates to a distance on that line then you have two linear systems where:

d(1) = start1 + rate1*time
d(2) = start2 + rate2*time

You are solving for time and start1 represents the proportion of the distance travelled with respect to the radius (i.e. the value of r*theta for a given theta) while the rate represents the speed rate in metres per second.

If the two distances are the same (i.e. d(1) = d(2)) then it means that both meet and this will allow you to solve for the unknown time value.

3. ## Re: Circular Track (First Meeting and Second Meeting)

Oh I got it now. Thanks