Exploring the Speed of a Positive Curved Time vs Distance Graph

[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?

 

[phinds]

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

 

[Stephen Tashi]

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?

 

[lekh2003]

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.

 

[sophiecentaur]

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.

 

[jbriggs444]

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.

 

[sophiecentaur]

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.

 

[jbriggs444]

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.

 

[sophiecentaur]

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.

 

[jbriggs444]

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.

 

[sophiecentaur]

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.