Sum of small displacement vectors.

In summary, the conversation discusses the concept of arclength and how it can be defined in relation to an object's motion in a 3D world. It is suggested that the arclength can be calculated by summing the length of displacement vectors over small time intervals. The idea is further supported by mentioning the Riemann sum for line integrals and the possibility of defining arclength using a differential mapping. However, it is noted that this approach may not work in all cases.
  • #1
STAii
333
1
Suppose we have an object.
The object is moving in a 3D world.
Now, let St1 - t2 denote the displacement vector from moment t1 to moment t2.
Now, let's say that t0 is the moment of the beginning of the motion, and (tf) is the last moment of movement.
We can split the time from t0 to tf into small bands, each x seconds long.
Now, let's add up the length of the vectors |St0 - t0+x| + |St0+x - t0+2x| + |St0+2x - t0+3x| + ... + |S(something) - tf| = Y
Now, it is obivous (at least for me) that if you make x smaller and smaller (x->0) then the value of Y will get nearer and nearer to the Distance passed by the object.
First of all, am i right ? Secondly (if so), how can it be prooved ?
Thanks !
 
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  • #2
That is (typically) how the arclength is defiend! What you described is just the riemann sum for the line integral

∫|ds|

The only other way of which I know that one could define the arclength of a path is if you can find a differential mapping from [0, t] to your curve such that the derivitive is always a unit vector. (intuitively this map would thus preserve length), then the arclength of your curve would be t.
 
  • #3
Of course this approach would not work in certain cases. e.g. tracing the mandlebrot curve, or a similar fractal.
 

1. What is the sum of small displacement vectors?

The sum of small displacement vectors is the total displacement of an object that has undergone a series of small movements. It is calculated by adding together all of the individual displacement vectors.

2. How do you calculate the sum of small displacement vectors?

To calculate the sum of small displacement vectors, you need to add together all of the individual displacement vectors. This can be done by adding the x and y components of each vector separately, using vector addition techniques.

3. Why is the sum of small displacement vectors important?

The sum of small displacement vectors is important because it helps us understand the overall movement of an object. By breaking down the individual movements into smaller vectors, we can accurately calculate the final displacement of the object.

4. Can the sum of small displacement vectors be negative?

Yes, the sum of small displacement vectors can be negative. This occurs when the individual vectors cancel each other out, resulting in a net displacement of zero or in the opposite direction of the initial displacement.

5. How is the sum of small displacement vectors represented?

The sum of small displacement vectors is typically represented by a single vector, known as the resultant vector. This vector has both magnitude and direction, and represents the final displacement of the object.

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