What Defines a 'Distance from...' Graph?

[Aeneas]

"Distance from..." graphs

At G.C.S.E level (England), there are still some questions knocking around about "distance from.." graphs as opposed to "distance traveled .." graphs. The first can go down as well as up, as the person returns to their starting place, while the second can only increase. These questions are often presented in the form of journeys, with horizontal parts of the graph presumably meant to represent rests, and sometimes questions about distance travelled, speed etc.

It seems to me, however, that these graphs are not defined. Is 'distance from' as the crow flies or distance traveled from in a straight line? If it is the first, the horizontal part of the line could represent the person traveling along an arc of a circle, the centre being their starting point. Also, you could tell nothing about distnce travelled. The person might be traveling along a spiral route, traveling a long distance but only getting gradually further away from their starting point.

Am I right in complaining about these "distance from.." graphs, or are there set conventions that define them?

Thanks in anticipation.

 

[disregardthat]

I may have completely misunderstood your question, but are you asking about how to find the distance of a given graph represented by a function from a point to another?

In that case it's easy to parameterise the function, differentiate it, and integrate the absolute value of it with the given limits.
$$L=\int^b_a |\vec{r}\prime(t)|dt$$

where $$\vec{r}(t)$$ is the parameterised function, and b are a are the upper and lower bound respectively

 

[Aeneas]

Sorry I didn't make it clearer. The kind of graphs I'm talking about are ones where 'Distance from a starting point' is plotted against time - as opposed to graphs where 'distance travelled' is plotted against time. The first can obviously go back down to y = 0 as you return home, the second can only increase irrespective of which direction you are moving in. In the case of the second type of graph, a horizontal portion of the line MUST indicate a rest. What I am arguing is that in the case of the first type, it does not need to do so, even though, in the case of exam questions, this is clearly the desired answer. I hope that's a bit clearer.

 

[disregardthat]

By a "rest" you mean that if the graph was describing a persons motion, he would be in rest at the horisontal point?

Indeed, if the distance traveled not rising, he cannot be moving.

If I understand your question, you wonder how a graph plotted against time can be horisontal even though a person is not in "rest", if the graph is describing distance from a starting point.

Ok, imagine a person walking in circles around an object, is the distance from the object increasing, decreasing or constant. How would you plot this on your graph? (describing the distance from object)

 

[Aeneas]

By a "rest" you mean that if the graph was describing a persons motion, he would be in rest at the horisontal point?

Indeed, if the distance traveled not rising, he cannot be moving.

If I understand your question, you wonder how a graph plotted against time can be horisontal even though a person is not in "rest", if the graph is describing distance from a starting point.

Ok, imagine a person walking in circles around an object, is the distance from the object increasing, decreasing or constant. How would you plot this on your graph? (describing the distance from object)

Exactly! It would be represented as a horizontal line, so you cannot tell whether the person is at rest or walking along an arc of a circle which has his starting point as its centre. This is part of my objection to these 'distance from' graphs, particularly as they are still cropping up in school courses and examinations. My question is - are these graphs as ambiguous as they seem, or are there set conventions surrounding them which define what they mean?

 

[disregardthat]

How they are defined? A graph is a graph, the graph you are talking about is describing the distance from an object. Is there more to it?