Rate of Change in Distance: Two persons are running along separate path....

[moremathlearner]

Could anyone please help me to solve the following question:

Two persons are running along separate path. The first is running at 2 pie km/hr along a circular track with a radius of 100 meter, while the second is running along the straight path, tangent to the first, at a rate of 8 km/hr. They both begin running from the point where the paths meet and at the same time.
How fast is the distance between them changing after an hour.

Thank you for your time.

 

[Deleted member 4993]

Could anyone please help me to solve the following question:

Two persons are running along separate path. The first is running at 2 pie km/hr along a circular track with a radius of 100 meter, while the second is running along the straight path, tangent to the first, at a rate of 8 km/hr. They both begin running from the point where the paths meet and at the same time.
How fast is the distance between them changing after an hour.

Thank you for your time.

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

 

[moremathlearner]

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

In one hour, first person completes 10 rounds. Therefore comes back to s original position. Now the other person travels 8 km from the staring point. So they are 8 km apart.
But want to know what is the rate of change of distance.

Thanks.

 

[mmm4444bot]

Two persons are running along separate [paths].

The first is running at 2 [pi] km/hr along a circular track with a radius of 100 meter, while the second is running along [a] straight path, tangent to the first, at a rate of 8 km/hr.

They both begin running from the point where the paths meet and at the same time.

How fast is the distance between them changing after an hour[?]

Is this exercise from a calculus course?

As they do not specify the relative direction of each runner at time zero, I had first assumed that the answer would be the same regardless. (Otherwise, how could one answer?) However, I get different results (using calculus), depending on the initial direction of each runner.

For example, let us position the straight path horizontally and tangent to the top of the circle. There are four cases:

\(\displaystyle \;\;\;\;\)Runner #1 travels clockwise, and Runner #2 travels east

\(\displaystyle \;\;\;\;\)Runner #1 travels clockwise, and Runner #2 travels west

\(\displaystyle \;\;\;\;\)Runner #1 travels counterclockwise, and Runner #2 travels east

\(\displaystyle \;\;\;\;\)Runner #1 travels counterclockwise, and Runner #2 travels west

I get a smaller result, when the initial directions are the same; I get a larger result, when the initial directions are opposite.

The attached graphs show this (t measured in hours). The red curve is the distance between the runners and the green curve is the rate at which the distance is changing.

I'm wondering whether I did something wrongly because the rate-of-change graph (in each case) shows a different value when runner #1 is halfway around the circle on their first lap than on the remaining nine laps. That seems odd, to me.

I had better spend some more time on this one! :cool:

 

[mmm4444bot]

Calculus is not needed, to answer this exercise, but calculus is how I confirmed my answers (after I realized there are two possibilities). I'm curious to know why my derivative function shows the same rate of change when Runner #1 is halfway around the circle on every lap except the first. I double-checked my work; I didn't find any issue.

I parameterized the positions of each runner.

For Runner #1 moving clockwise around a circle with radius r, I used:

X(t) = r*cos(θ₀ - ω*t)

Y(t) = r*sin(θ₀ - ω*t)

where the initial angle θ₀ provides the starting location and ω is the angular speed (ratio of angle swept to time elapsed).

The runners are at (0, 1/10) at time zero, so θ₀ = Pi/2.

The circumference of the circle is Pi/5 km. In one hour, Runner #1 travels 2*Pi km.

2*Pi / (Pi/5) = 10 laps in one hour, so the angle swept is 10*2*Pi and ω = 20*Pi rad per hour.

Substituting and simplifying, I got:

X(t) = 1/10*sin(20*Pi*t)

Y(t) = 1/10*cos(20*Pi*t)

For Runner #2 moving east, I used:

x(t) = 8*t

y(t) = 1/10

The distance formula yields a function for the distance between the runners at time t.

f(t) = sqrt( [X(t) - x(t)]^2 + [Y(t) - y(t)]^2 )

Does this look good, so far?