# How to calculate average velocity?

## Main Question or Discussion Point

Hello,

Suppose given a sample data of particle travelling with following co-ordinates:

x | y
[(2, 2),
(4, 2),
(4, 3),
(4, 4),
(1, 3),
(1, 1),
(4, 4),
(4, 1),
(0, 4),
(4, 2)]

we want to calculate the Average velocity. How do we find it?
If we use difference between initial and last wont give the average as the particle might have traveled a lot prior to that.

I have a data of around 1000 points where the particle traveled. And thus would like to find the same. Can you help me solving it.
Thanks.

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What you have listed are ordered pairs (x,y) which to me means two spatial coordinates. If that is true then there is no way to determine the average velocity. You need time intervals.

What you have listed are ordered pairs (x,y) which to me means two spatial coordinates. If that is true then there is no way to determine the average velocity. You need time intervals.
Oh yes, they are ordered pairs but if we consider time interval to be 1 sec (say or 1 time unit ), how would I do it?

Student100
Gold Member
Then take you x column and rewrite it as:

x | t
2 1
4 2
4 3
4 4
1 5
1 6
4 7
4 8
0 9
4 10

Then do something similar with y. It basically makes x/y dependent on t, so you can calculate average velocity rather straight forwardly. For 1000 data points you're going to want to use excel or Matlab.

DrGreg
Gold Member
If we use difference between initial and last wont give the average as the particle might have traveled a lot prior to that.
Actually, that is the definition of average velocity (after you divide by time, of course). Remember, velocity is a vector, and if the velocity keeps changing direction, a lot of the contributing velocities could cancel each other out.

However, if you really want average speed rather than velocity, then calculate the speed along each segment and average them.

Then take you x column and rewrite it as:

x | t
2 1
4 2
4 3
4 4
1 5
1 6
4 7
4 8
0 9
4 10

Then do something similar with y. It basically makes x/y dependent on t, so you can calculate average velocity rather straight forwardly. For 1000 data points you're going to want to use excel or Matlab.
Sorry, I didn't get you.
By, making dependent on t, do you mean to do: ∑x/∑t ?
I will use python to calculate the average velocity. But, I cannot look straight forward.

Actually, that is the definition of average velocity (after you divide by time, of course). Remember, velocity is a vector, and if the velocity keeps changing direction, a lot of the contributing velocities could cancel each other out.

However, if you really want average speed rather than velocity, then calculate the speed along each segment and average them.
Thanks, Dr. Greg.
I don't want average speed but I would like to have average velocity. I have the data of average speed. More like Σ(x2+y2)-1/t

Student100
Gold Member
Sorry, I didn't get you.
By, making dependent on t, do you mean to do: ∑x/∑t ?
I will use python to calculate the average velocity. But, I cannot look straight forward.
Let me make sure you're trying to calculate what I think you're wanting to calculate. Are you trying to do a statistical analysis on this data (find multiple average velocities over the set and compare them)? If not you don't actually have to use anything if you just want average velocity over the whole set. You could then just use the last and first data points, with the known time between the two.

Motion in two dimension (your ordered pairs) is independent of one another. You need the data to be in terms of parametric equations to calculate average velocity. That's what I mean by making x/y dependent on t.

Let me make sure you're trying to calculate what I think you're wanting to calculate. Are you trying to do a statistical analysis on this data (find multiple average velocities over the set and compare them)? If not you don't actually have to use anything if you just want average velocity over the whole set. You could then just use the last and first data points, with the known time between the two.

Motion in two dimension (your ordered pairs) is independent of one another. You need the data to be in terms of parametric equations to calculate average velocity. That's what I mean by making x/y dependent on t.
Hey, Thanks.
I tried making it clear that I wouldn't want to calculate average being first and last data point. I would like to calculate translational energy of the particle. (I am stuck with Langevin equation.)For that I would require average velocity.

DrGreg
Gold Member
I tried making it clear that I wouldn't want to calculate average being first and last data point.
Then whatever you want, it isn't the average velocity.
I would like to calculate translational energy of the particle.
Do you mean average kinetic energy? Then calculate the kinetic energy between each pair of points, and average.

Student100
Gold Member
Hey, Thanks.
I tried making it clear that I wouldn't want to calculate average being first and last data point. I would like to calculate translational energy of the particle. (I am stuck with Langevin equation.)For that I would require average velocity.
Transitional kinetic energy?

I'm still a bit confazzled by what you mean "I don't want the average being the first and last data point." That's basically the definition of the average velocity over a given time period. Now, I understand if you mean you want to calculate multiple averages from your data set, which is certainly possible and perhaps interesting. In that case you have your work cut out for you, and will need to write a python script to look at as many of them as you can.

I'm still a bit confazzled by what you mean "I don't want the average being the first and last data point."
By that I meant, I don't want the average of (Xi-X1)/2. Is it bit more clear?

Transitional kinetic energy?
Ah yes. Exactly, I would like to find out Kinetic energy. 1/2*m*v2.

For instance, if the particle has traveled a lot in the time frame for say 1-1000 seconds.

Now, I understand if you mean you want to calculate multiple averages from your data set
I am not sure how do I use the term Multiple averages. Do you mean, x2-x1,y2-y1 ?
In that case you have your work cut out for you, and will need to write a python script to look at as many of them as you can.
I can iterate throughout the condition. Programming is not a problem.

I am finding it difficult to find out mathematically.
So, in the Kinetic energy, how do I find the v2 which is the average velocity of the particle travelling. ?

Student100
Gold Member
By that I meant, I don't want the average of (Xi-X1)/2. Is it bit more clear?
You wouldn't use (Xf-Xi)/2, you would use (xf-xi)/(tf-ti). That's basically average velocity.

Ah yes. Exactly, I would like to find out Kinetic energy. 1/2*m*v2.

For instance, if the particle has traveled a lot in the time frame for say 1-1000 seconds.
Then you could just use 1/2m*(insert average velocity calculated in the above example)^2

I am not sure how do I use the term Multiple averages. Do you mean, x2-x1,y2-y1 ?
No I mean you take the average velocity between time points of x. With a data set of 1000 points you could calculate quite a few of averages. I.e average over 1000 seconds, the average over 40 seconds, or the average from t=3 to t= 57 if you want.

I can iterate throughout the condition. Programming is not a problem.

I am finding it difficult to find out mathematically.
So, in the Kinetic energy, how do I find the v2 which is the average velocity of the particle travelling. ?
Exactly like above.

• wokay
No I mean you take the average velocity between time points of x. With a data set of 1000 points you could calculate quite a few of averages. I.e average over 1000 seconds, the average over 40 seconds, or the average from t=3 to t= 57 if you want.
Don't you think, this will be like calculating mean square displacement?

Student100
Gold Member
Don't you think, this will be like calculating mean square displacement?
Not unless you also care about probability.

Not unless you also care about probability.
Thank you Student100 for your help. I am not yet fully satisfied with the methodology, but seems fair to a point that you consider few averages. But the few seems to be little concerning.

Now, considering this to be valid,
No I mean you take the average velocity between time points of x. With a data set of 1000 points you could calculate quite a few of averages. I.e average over 1000 seconds, the average over 40 seconds, or the average from t=3 to t= 57 if you want.
I take the position vector r = sqrt(x^2+y^2) and take few averages, seems fair enough?

Also, I would appreciate if the above methodology seems convincing for few others.

Thank you.