# Exploring the Speed of a Positive Curved Time vs Distance Graph

• TheOGBacon
In summary, the conversation discusses the concept of finding a single speed for a positively curved time vs distance graph, and the difficulty in doing so due to changing slopes and the presence of acceleration. It is suggested that one could calculate an average speed between two points or use calculus to find the instantaneous speed at a given point. The conversation also touches on the idea of using a curve of best fit to clean up the data and the relationship between upward curvature and acceleration.
TheOGBacon
If there was a positive curved time vs distance graph going upwards and to the right, would there be a speed for this graph even if all the points do not have a slope in common?

TheOGBacon said:
If there was a positive curved time vs distance graph going upwards and to the right, would there be a speed for this graph even if all the points do not have a slope in common?
You could apply a best-fit algorithm to get a reasonable average speed function, assuming that the points are now wildly off of a trendline

TheOGBacon said:
If there was a positive curved time vs distance graph going upwards and to the right, would there be a speed for this graph even if all the points do not have a slope in common?

There would not be a single number that defines the speed the whole graph. One could calculate an average speed between two given points in time. One could also compute a speed "at" a particular instant of time. Calculating speed at an instant in time gets into the concepts of Calculus. What math have you studied?

TheOGBacon said:
If there was a positive curved time vs distance graph going upwards and to the right, would there be a speed for this graph even if all the points do not have a slope in common?
You will never be able to find the most accurate speed, since I am assuming the slope is always changing.

You can however use Calculus and the definition of the derivative to find the slopes and averages across increasingly small distances and eventually instantaneously.

TheOGBacon said:
If there was a positive curved time vs distance graph going upwards and to the right, would there be a speed for this graph even if all the points do not have a slope in common?
If a distance / time graph has a curve to it then that means there is acceleration. The slope of the graph at any point is the instantaneous speed. Your points are presumably, measured values and it's likely that the scatter is due to either simple measurement errors of some variable in the dirving force / frictions forces.
What you do with the data will depend on what you want out of it. As @Stephen Tashi says, you can find the average speed from total distance / total time but you can clean up the data by drawing (by eye) as smooth a curve as possible that passes through or between the points as near as possible. (Like driving a fast car through a set of traffic cones which haven't be placed very accurately - you can't steer to hit them all but you do your best). There are maths procedures that will give you a better curve of best fit but by-eye can be pretty good. If you draw a tangent to the curve at any point then the instantaneous speed will be given by the slope at that point. Measuring the speed along the journey in that way will tell you how the acceleration changes. IF the slope is always 'upwards, the acceleration is increasing over the journey.

sophiecentaur said:
IF the slope is always 'upwards, the acceleration is increasing over the journey.
If the slope of what is upwards?
If the slope of the distance/time graph is upwards, all that tells you is that the speed is positive.
If the slope of the speed/time graph is upwards, all that tells you is that the [tangential] acceleration is positive.
If the slope of the acceleration/time graph is upwards, that tells you that acceleration is increasing.

jbriggs444 said:
If the slope of what is upwards?
If the slope of the distance/time graph is upwards, all that tells you is that the speed is positive.
If the slope of the speed/time graph is upwards, all that tells you is that the [tangential] acceleration is positive.
If the slope of the acceleration/time graph is upwards, that tells you that acceleration is increasing.
The OP describes an upwards curve, as I read it. If it is curved then there is acceleration. Of course, a picture of the graph with properly labelled axes would have helped.

sophiecentaur said:
The OP describes an upwards curve, as I read it. If it is curved then there is acceleration. Of course, a picture of the graph with properly labelled axes would have helped.
Fair enough. Though an upward curve to the distance/time graph indicates positive acceleration, not increasing acceleration.

jbriggs444 said:
Fair enough. Though an upward curve to the distance/time graph indicates positive acceleration, not increasing acceleration.
That would depend upon the derivative of the curvature of that graph. Second year and not first year work, I think.

sophiecentaur said:
That would depend upon the derivative of the curvature of that graph.
For any reasonable definition of curvature I can come up with, it [upward curvature] would be associated with increasing speed and positive acceleration, not increasing acceleration.

I reckon I could draw two s/ t graphs which 'curve upwards', one with increasing and one with decreasing acceleration and you would not be able to eyeball which was which - unless you could see them side by side. We are talking about a second derivative and the brain is no too good with that.

jbriggs444

## 1. What is a positive curved time vs distance graph?

A positive curved time vs distance graph is a type of graph that shows the relationship between time and distance traveled. It is called "positive" because the line on the graph slopes upward, indicating that the distance is increasing as time goes by. The curve represents the acceleration of an object, with steeper curves indicating faster acceleration.

## 2. How is the speed calculated from a positive curved time vs distance graph?

The speed can be calculated by finding the slope of the line at any given point on the graph. This slope represents the object's velocity at that specific time. The steeper the slope, the faster the object is moving. To find the average speed, you can divide the total distance traveled by the total time taken.

## 3. How does a positive curved time vs distance graph differ from a straight line graph?

A positive curved time vs distance graph differs from a straight line graph in that the slope of the line is constantly changing. In a straight line graph, the slope remains the same throughout, indicating a constant speed. In a positive curved graph, the slope changes as the object accelerates or decelerates.

## 4. What does the area under the curve represent in a positive curved time vs distance graph?

The area under the curve represents the displacement or total distance traveled by the object. This is because the curve represents the object's velocity, and the area under the curve is the distance traveled during that time interval. The larger the area, the greater the displacement or distance traveled.

## 5. How does a positive curved time vs distance graph relate to real-life scenarios?

A positive curved time vs distance graph can be used to represent various real-life scenarios, such as a car accelerating and decelerating, an airplane taking off and landing, or a person running and then stopping. It can also be used to analyze the speed and acceleration of objects in sports, such as a ball being thrown or a car in a race.

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