What is the acceleration of the runner?

bee-gee

1. The problem statement, all variables and given/known data
how do i work out the acceleration from a curved line on a distance-time graph?
2. Relevant equations


3. The attempt at a solution

tiny-tim

hi bee-gee! welcome to pf!
you obviously know how to find the speed (from the slope of area under the graph)

i suppose you could use that to draw a speed-time graph, and then measure the slope of area under that

alternatively, if you can measure the radius of curvature, r, and if you know about centripetal acceleration, then you can do it by imagining that the graph represents a rocket rising vertically …

now put the rocket in a constant horizontal wind, so that sideways distance is proportional to time: if you get the constant right, the vertical-horizontal distance-distance graph will look exactly the same as the original distance-time graph, and you know the total acceleration is purely vertical

sjb-2812

Isn't speed the slope of a distance / time graph?

tiny-tim

oops!

i typed too quickly!

thanks, i'll edit it

HallsofIvy

Yes. I'm sure tiny-tim just mis-spoke. sjb-2812, a cute method I was taught, in highschool, for finding tangents, and so derivatives, from a graph, used a small mirror. Hold the mirror on the graph at the point at which you want the tangent. Rotate the mirror about the vertical axis until the graph seems to go smoothly into its reflection in the mirror. Use the mirror as a straight edge to draw a line perpendicular to the curve. Now turn the mirror to draw a line perpendicular to that line and repeat the process to draw a line perpendicular to the original perpendicular and so tangent to the curve. From the values on the graph of the endpoints of that tangent line, you can find the slope of the tangent line and so the derivative of the function.

To find the acceleration, the second derivative of distance, you would need to do that in a number of different places to get a rough graph of "velocity versus time" and then repeat the process with that graph.

The simplest thing to do is to approximate the speed, at a number of points using the difference quotient of a number of near by points. Then, again, repeat the process to find the second derivative.