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

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In summary, "distance from.." graphs and "distance traveled.." graphs are often used interchangeably in school courses and examinations, causing confusion. However, "distance from.." graphs can be ambiguous as they do not specify if the distance is measured as the crow flies or in a straight line from the starting point. There are no set conventions surrounding these graphs, leading to different interpretations and making them unreliable for measurement.
  • #1
Aeneas
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"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.
 
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  • #2
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.
[tex]L=\int^b_a |\vec{r}\prime(t)|dt[/tex]

where [tex]\vec{r}(t)[/tex] is the parameterised function, and b are a are the upper and lower bound respectively
 
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  • #3
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.
 
  • #4
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)
 
  • #5
Jarle said:
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?
 
  • #6
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?
 

What is distance from a graph?

Distance from a graph refers to the measurement of the space between two points on a graph. It is commonly used in mathematics and science to calculate the distance between two data points on a graph.

How do you calculate distance from a graph?

The distance between two points on a graph can be calculated using the Pythagorean theorem. This involves finding the difference in the x-coordinate and y-coordinate of the two points, squaring each difference, adding the squares together, and then taking the square root of the sum.

What is the unit of measurement for distance on a graph?

The unit of measurement for distance on a graph depends on the scale of the graph. It could be in centimeters, meters, kilometers, or any other unit of length. It is important to pay attention to the units when calculating and interpreting distances on a graph.

Why is distance from a graph important?

Distance from a graph is important because it allows us to quantify the relationship between two variables. It can help us analyze patterns and trends, make predictions, and draw conclusions about the data being represented on the graph.

How can distance from a graph be used in real-life applications?

Distance from a graph can be used in many real-life applications, such as calculating the distance between two cities on a map, predicting the growth of a population over time, or determining the speed of an object based on its position at different points in time. It is also commonly used in fields like engineering, economics, and physics to solve problems and make informed decisions.

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