# Thread: Track runner coordinates problem

1. ## Track runner coordinates problem

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost point. They both run counterclockwise. Alexandra runs at 4 meters per second, and will take exactly 3 minutes to catch up to Charlie.

Impose a coordinate system with units in meters where the origin is at the center of the circular track, and give the x- and y-coordinates of Charlie after one minute of running.

I really don't get how to even start this problem. I've got my coordinate system set up, and I thiiiink the next step is to find Charlie's angular speed? but how? step-by-step would be super helpful on this problem. I've received some promptings by tutors but I've reached a mental block.

2. ## Re: Track runner coordinates problem

Have you learned Polar coordinates yet? It is easiest to write everything up in Polar coordinates and then convert to rectangular coordinates. So, in Polar coordinates, both Charlie and Alexandria have constant radius $60$, so their positions $(r,\theta)$ are $(60,\theta_A(t))$ and $(60,\theta_C(t))$. The circumference of the track is $2\pi r = 120\pi$ meters. So, in $t$ seconds, Alexandria travels $\dfrac{4t}{120\pi}$ of the full track.
To find the measure in radians, multiply by $2\pi$, so $\theta_A(t) = \theta_A(0) + \dfrac{t}{15}$ radians. Since her starting angle is $\dfrac{\pi}{2}$, you have $\theta_A(t) = \dfrac{\pi}{2}+\dfrac{t}{15}$ radians.
To find the measure in degrees, multiply by $360^\circ$, so $\theta_A(t) = \theta_A(0) + 12t$ degrees. Since her starting angle is $90^\circ$, you have $\theta_A(t) = 90+12t$ degrees.

Now, at three minutes, $t = 180$, so $(r,\theta_A(180)) = (r,\theta_C(180))$. Assuming Charlie's speed is constant, $\theta_C(t) = \theta_C(0) + t k$ where $k$ is Charie's angular speed in either radians per second or degrees per second. Since Charlie's initial position is $(-60,0)$, you know $\theta_C(0) = \pi$ if you want the measure in radians and $\theta_C(0) = 180^\circ$ if you are using degrees.

Then, in radians: $\theta_A(180) = \dfrac{\pi}{2}+\dfrac{180}{15} = \pi + 180k = \theta_C(180)$. Solving for $k$ gives $k = \dfrac{24-\pi}{360}$ radians per second. So, Charlie's position (with theta in radians) is $\left(60,\pi + \dfrac{24-\pi}{360}t\right)$.

In degrees: $\theta_A(180) = 90+12(180) = 180+180k = \theta_C(180)$. Solving for $k$ gives $k = \dfrac{23}{2}$ degrees per second. So, Charlies position (with theta in degrees) is $\left(60,\left(180+\dfrac{23}{2}t\right)^\circ \right)$.

To go from polar coordinates to rectangular coordinates, you just have $x = r\cos \theta$ and $y = r\sin \theta$. So, Charlie's position at one minute is given by $x = 60\cos\left(\pi + \dfrac{24-\pi}{6}\right), y = 60\sin\left(\pi+\dfrac{24-\pi}{6}\right)$ (using theta in radians) or $x = 60\cos\left(180+23\cdot 30)^\circ, y = 60\sin\left(180+23\cdot 30)^\circ$ (using theta in degrees). This gives his position after one minute to be approximately $(56.6684,19.7153)$.

3. ## Re: Track runner coordinates problem

Originally Posted by UWstudent
Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost point. They both run counterclockwise. Alexandra runs at 4 meters per second, and will take exactly 3 minutes to catch up to Charlie.

Impose a coordinate system with units in meters where the origin is at the center of the circular track, and give the x- and y-coordinates of Charlie after one minute of running.

I really don't get how to even start this problem. I've got my coordinate system set up, and I thiiiink the next step is to find Charlie's angular speed? but how? step-by-step would be super helpful on this problem. I've received some promptings by tutors but I've reached a mental block.
It would help if you told us what your coordinate system is. You understand that the answer depends on that don't you?
Personally, I would set up a rectangular coordinate system with the origin at the center of the circular track, positive x-axis to the east, positive y-axis to the north, but I can think of many others that would work.

So Charlie starts at (-60, 0) and Alexandra starts at (60, 0). Alexandra runs at 4 meters per second and the entire track has length "pi times diameter", $60\pi$ meters. At 4 meters per second, it will take Alexandra 60/4= 15 times pi, $15\pi$ seconds. That is the same as saying she runs $2\pi$ radians in $15\pi$ seconds or $\frac{2}{15}$ radians per second.

The angular difference between Charlie and Alexandra is initially $3\pi/2$ radians and it takes Charlie 3 minutes= 180 seconds to make that up so Charlie's angular speed is $((3\pi)/2)/180= \pi/120$ radians per second faster than Alexandria- Charlie's angular speed is $\frac{2}{15}+ \frac{\pi}{180}$ radians per second.

4. ## Re: Track runner coordinates problem

Originally Posted by HallsofIvy
the entire track has length "pi times diameter", $60\pi$ meters.
Since radius is 60 meters, diameter is 120 meters.

Originally Posted by HallsofIvy
The angular difference between Charlie and Alexandra is initially $3\pi/2$ radians and it takes Charlie 3 minutes= 180 seconds to make that up so Charlie's angular speed is $((3\pi)/2)/180= \pi/120$ radians per second faster than Alexandria- Charlie's angular speed is $\frac{2}{15}+ \frac{\pi}{180}$ radians per second.
The OP stated that Alexandria catches up to Charlie, so I read that as Alexandria is running faster than Charlie.

5. ## Re: Track runner coordinates problem

@SlipEternal

I have never heard of the polar coordinate system. I looked it up, it looks like we've been learning from the Cartesian system. However, your explanation was thorough and I think I could replicate the ideas if I see a similar problem!! It was kind of a revelation to see the 4t/120pi bit. The cool thing, but also the hard thing, about circles is that they can be manipulated in so many different ways! Anyways, thank you so much. This really helped a lot!

6. ## Re: Track runner coordinates problem

@HallsofIvy

I didn't realize there was more than one coordinate system! Yes, I set up a rectangular system. I do see now how the problem depends on the coordinate system.