Finding the Velocity from Acceleration as a function of position

  • #1
Hey, I've just found this forum and it looks like a great place for science-minded people. :smile:

I'm having trouble finding the velocity from a graph of acceleration as a function of position.

Homework Statement



http://www.adam.com.au/shiloh/Graph.png

The Attempt at a Solution



I've assumed that you can find the integral of each triangle and that this would be the velocity as a function of position. I used A = 0.5*x*a to find these areas. I calculated area 1 to be 2.925 m/s and area 2 to be -2.0475 m/s. I then summed the areas with the initial velocity given.

My final answer was 4.8775 m/s but when I try to submit this answer, its wrong. Could someone please point me in the right direction so I can solve this?

Thanks!
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,836
251
Welcome to PF!

I'm having trouble finding the velocity from a graph of acceleration as a function of position.

I've assumed that you can find the integral of each triangle and that this would be the velocity as a function of position. I used A = 0.5*x*a to find these areas. I calculated area 1 to be 2.925 m/s and area 2 to be -2.0475 m/s. I then summed the areas with the initial velocity given.

Hi astronutties! Welcome to PF! :smile:

I stared at this for several minutes before I noticed what had happened …

your method is correct :smile: … but only if the horizontal axis is time! :wink:
 
  • #3
Thanks for the welcome tiny-tim!

I thought the same process could be used for acceleration vs position as for an acceleration vs time graph. I did manage to solve it in the end but I wasn't entirely sure what I did...just a lot crunching through different methods until one worked. :wink:

In case anyone was interested, the answer was 4.198 m/s.

Thanks again for the help.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,847
965
a= dv/dt= (dv/dx)(dx/dt)= v dv/dx, by the chain rule. Knowing the acceleration as a function of time means you have v dv/dx= a(x) or v dv= a(x)dx. You can integrate both sides to get v as a function of x.
 
  • #5
Hi HallsofIvy,

That's what I thought I was doing. By finding the area under the graph, I thought I was finding the integral of a(x) which would be v(x) when rearranged as you have just said.
 
  • #6
D H
Staff Emeritus
Science Advisor
Insights Author
15,415
687
That's what I thought I was doing. By finding the area under the graph, I thought I was finding the integral of a(x) which would be v(x) when rearranged as you have just said.
That is not what Halls said. He arrived at [itex]v dv= a(x)dx[/itex], which when integrated from position [itex]x=0[/itex] to position [itex]x=3[/itex] yields

[tex]\frac 1 2 (v(x=3)^2 - v(x=0)^2) = \int_{x=0}^{x=3} a(x) dx [/tex]

You can solve for the velocity at position [itex]x=3[/itex] using the above.

You seem to have missed Tiny Tim's point: The graph is a function of distance, not time. While the integral of acceleration with respect to time is velocity, the integral of acceleration with respect to distance is something different. Just look at the units: Acceleration times time has units of velocity, but acceleration times distance has units of length2/time2, which is the same units as velocity squared.
 
  • #7
tiny-tim
Science Advisor
Homework Helper
25,836
251
Hi astronutties! :smile:
Hi HallsofIvy,

That's what I thought I was doing. By finding the area under the graph, I thought I was finding the integral of a(x) which would be v(x) when rearranged as you have just said.

No … the area under the curve is ∫vdv, which is … ? :smile:
 
  • #8
No … the area under the curve is ∫vdv, which is … ? :smile:

Oh I think I see where you are all getting at now. So the area under the graph is equivalent to the displacement? That puts everything into perspective now.

Thanks for your help!
 
  • #9
Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
12,133
160
Oh I think I see where you are all getting at now. So the area under the graph is equivalent to the displacement? That puts everything into perspective now.

Thanks for your help!

No, it's not displacement. The area under the curve is the same as

[tex]
\int v \ dv
[/tex]

Do that integral, and you'll be nearly done.
 
Last edited:

Related Threads on Finding the Velocity from Acceleration as a function of position

Replies
1
Views
2K
  • Last Post
Replies
1
Views
6K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
5
Views
13K
Replies
13
Views
2K
Replies
5
Views
197
Replies
12
Views
2K
Top