MHB Solve Distance Problem A & B: 6 & 10 Min Circular Mile Track

  • Thread starter Thread starter bergausstein
  • Start date Start date
AI Thread Summary
A and B run around a circular mile track in 6 and 10 minutes, respectively. To find when they pass each other, the approach involves using their speeds and the concept of distance. The faster runner will lap the slower runner when the distance covered by the faster runner equals the distance covered by the slower runner plus one circumference of the track. By setting up the equation based on their velocities and time, the solution can be derived. The discussion emphasizes simplifying the problem for better understanding, especially for those new to algebra.
bergausstein
Messages
191
Reaction score
0
A and B can run around a circular mile track in 6 and 10 minutes respectively. If they start at the same instant from the same place, in how many minutes will they pass each other if they run around the track (a) in the same direction, (6) in opposite directions?

can you help solve the first part of the question? thanks!
 
Mathematics news on Phys.org
I think I would first observe that the size of the track is irrelevant, and we can let the radius of the track be 1 unit. Then, I would describe the position of the runners parametrically, with time = $t$, as the parameter, measured in minutes. Let the runners begin at (1,0) and move in a counter-clockwise direction. Then their positions can be given by:

$$x(t)=\cos\left(\frac{2\pi}{T}t \right)$$

$$y(t)=\sin\left(\frac{2\pi}{T}t \right)$$

where $T$ is the period of their motion, i.e., the time (in minutes) it takes for them to complete one circuit of the track.

Then equate the respective coordinates of both runners, and take the first positive solution for $t$.
 
MarkFl, I appreciate what you posted above. But it seems that it isn't in the realm of what I can comprehend at this point in time since I'm just beginning to learn algebra. can you show me the simple approach to this problem? thanks!
 
Yes, now that I think about it more, there is a much simpler approach. :D

Let $C$ be the circumference of the track, and using distance = rate times time, and subscripting the faster runner's parameters with a 1 and the slower runner with a 2. We may use the fact that when the faster runner laps the slower runner, his distance ran will be one more circumference than the slower runner, and write:

$$d_1=d_2+C$$

Use $d=vt$:

$$v_1t=v_2t+C$$

What are the velocities of the two runners?
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top