Could there be two possible answers in the speed equation

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Raph27

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One day, two friends went their separate way, one moving to the north while the other to the east. Friend A is going to the north @ 5 ft/second while friend B is going east @ 1 ft/ second. How fast wd they be going from each other after 5 seconds. The answer is i blv 5.099 ft/s. My question is, could there be 2 answers, one is the speed based from the route they each took (right angle distance or the L road for which they took, while the other is the speed after 5 seconds in a straight line. (Since the direction was not an opposing straight line, the default as per mathematical principle is i blv the final position/destination after five seconds which is a straight line from 2 points, not the original route or road they took. Sorry but i was perplexed when this question was brought to my attention by someone, since i am not a mathematician and this is also a math problem of my nephew. :)

Thanks,

Raph
 
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The distance between them is the length of a straight line segment that connects them, right? This distance is a function of time. The speed is the first derivative (also a function of time). Plug in 5 sec to get the answer, which will be unique.
 
Raph27 said:
One day, two friends went their separate way, one moving to the north while the other to the east. Friend A is going to the north @ 5 ft/second while friend B is going east @ 1 ft/ second. How fast wd they be going from each other after 5 seconds. The answer is i blv 5.099 ft/s. My question is, could there be 2 answers, one is the speed based from the route they each took (right angle distance or the L road for which they took, while the other is the speed after 5 seconds in a straight line. (Since the direction was not an opposing straight line, the default as per mathematical principle is i blv the final position/destination after five seconds which is a straight line from 2 points, not the original route or road they took. Sorry but i was perplexed when this question was brought to my attention by someone, since i am not a mathematician and this is also a math problem of my nephew. :)

Thanks,

Raph
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The only way there could be two CORRECT answers would be if there were two VALID interpretations of what the problem means.

It clearly specifies what each is doing (running along a straight-line path in a given direction). The distance between them is a straight-line distance, and that is increasing at a rate determined by how each is moving.

I'm not sure what you are thinking, but in saying, "the final position/destination after five seconds which is a straight line from 2 points", you may be thinking of the average rate rather than the instantaneous rate, which is what they are asking for. The latter depends not only on where they are and how long it took to get there, but on how they are moving at that moment.

(On the other hand, this is a particularly simple problem that doesn't even need calculus, as the distance at time t turns out to be \(\sqrt{26}t\), so the rate is just \(\sqrt{26}\approx 5.099\).)
 
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Thanks for the answer guys. I apologize but i was just perplexed by the comment that the speed could be different, based on the image below, one is if we are to compute the speed based on the route taken, B and A (distance z, x and y). And the speed if we are to based it from a straight line, (distance between the two red dots, Z and Y). In this case, we disregard the actual route used but just measure the distance between the two friends after 5 seconds.

The answer is indeed as explained by Dr Peterson and Lev888. :)

My friend is a darn philo major and he is putting a twist on the question. Maybe the question should identify the speed based on route taken or will it be the same (the answer) if we will just consider the speed based on a straight line between the two friends after 5 seconds. X is the starting point from which the two friends separated. The distance considering the road taken is different as compared to the distance in a straight line from where they are after 5 seconds.

Thanks in advance for your opinion. :)
 

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Raph27 said:
Thanks for the answer guys. I apologize but i was just perplexed by the comment that the speed could be different, based on the image below, one is if we are to compute the speed based on the route taken, B and A (distance z, x and y). And the speed if we are to based it from a straight line, (distance between the two red dots, Z and Y). In this case, we disregard the actual route used but just measure the distance between the two friends after 5 seconds.

The answer is indeed as explained by Dr Peterson and Lev888. :)

My friend is a darn philo major and he is putting a twist on the question. Maybe the question should identify the speed based on route taken or will it be the same (the answer) if we will just consider the speed based on a straight line between the two friends after 5 seconds. X is the starting point from which the two friends separated. The distance considering the road taken is different as compared to the distance in a straight line from where they are after 5 seconds.

Thanks in advance for your opinion. :)
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If the problem was about distance _along_ a route, I think it would've been specified.
 
Raph27 said:
My friend is a darn philo major and he is putting a twist on the question. Maybe the question should identify the speed based on route taken or will it be the same (the answer) if we will just consider the speed based on a straight line between the two friends after 5 seconds. X is the starting point from which the two friends separated. The distance considering the road taken is different as compared to the distance in a straight line from where they are after 5 seconds.
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I thought philosophers understood the importance of clear communication, and of paying attention to exactly what someone says. At the very least, your friend should recognize the difference between a problem with two answers, and a problem that is stated ambiguously.

One small problem we have here is that you haven't shown us the exact wording, which makes it harder to answer your friend fully. I'm sure it's stated more clearly than your paraphrase.

The problem, even in your paraphrase, tells you clearly what to "consider". What speed is it asking about? This: "How fast would they be going from each other after 5 seconds?" That would mean, at what rate is the distance between them increasing at that time? Okay, I suppose you could ask how fast the distance along the road is increasing; and outside of the context of a calculus problem, that might be sensible. But within this context, you know what they mean: They mean whatever makes it an interesting calculus problem!

How about showing us the actual problem, so we can blame its author is there is really so much ambiguity?
 
I have been taught:

Speed (without any other adjective) is the magnitude of the velocity vector. Sometimes it is called "instantaneous" speed.

Average (constant) speed is the rate of change of distance. So if a vehicle covers a semicircular path (radius = r) in time 't',

the Average (constant) speed is = (distance)/time = (π*r)/t​
​
The equivalent constant velocity = (displacement)/t = (2*r)/t i (i is the assumed unit vector along the diameter)​
 
@Raph27
Your friend is clever! Although, I think what he is saying is possibly tongue-in-cheek. He has made the point (knowingly or not), that in mathematics there are different ways of measuring distances. One way of measuring the distance between two points is the common, straight-line distance, which is of course what any question will be asking about. The 'Euclidean metric' measures that distance. However there is another way of measuring the distance between two points, called the 'Taxicab metric', which does what your friend is describing. (If curious, you might google them. However the taxicab metric can safely be ignored!)
 
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lex said:
@Raph27
Your friend is clever! Although, I think what he is saying is possibly tongue-in-cheek. He has made the point (knowingly or not), that in mathematics there are different ways of measuring distances. One way of measuring the distance between two points is the common, straight-line distance, which is of course what any question will be asking about. The 'Euclidean metric' measures that distance. However there is another way of measuring the distance between two points, called the 'Taxicab metric', which does what your friend is describing. (If curious, you might google them. However the taxicab metric can safely be ignored!)
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yes i agree with you. You nailed the answer haha thanks a lot Lex and Dr. Peterson. Googled it and it was quite enlightening..
 
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