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## Homework Statement

A particle of dust is bombarded by air molecules and follows a zigzag path at constant speed v.

(a)Assuming each step has a length d, find the distance traveled by the dust particle in time t.

(b)What is the length of the displacement vector after N steps where N is large? Assume that each step is taken in a random direction on the plane. (This problem assumes you are familiar with the scalar product of two vectors.

## Homework Equations

v

^{2}=(v

_{1}

^{2}+v

_{2}

^{2}+...+v

_{N}

^{2})/N

## The Attempt at a Solution

The answer to part (a) seemed quite obvious: vt.

However, part (b) isn't so easy. I've been able to get the answer d[tex]\sqrt{N}[/tex] by doing vt=t x sqrt[(v

_{1}

^{2}+v

_{2}

^{2}+...+v

_{N}

^{2})/N] but this is getting the answer for the sake of it. My method doesn't actually make any sense because it can cancel out to nothing. Please help!

(I believe this is called Brownian motion?) (the thing I don't get is how do I use dot product in this instance?)

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