# Calculating distance, angle bet. velocity and acceleration

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1. Jul 18, 2017

### Pushoam

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
A)

s = √[(x2) + (y2 ) ]= a√[2(1- cos (ωt) ) ]|t= Γ

The book says, s = aωΓ

What I can do is ,
For very small Γ i.e. ωΓ<<1 , cos (ωΓ) ≈ 1 - {(ωΓ)2}/2

Then , I get,
s = aωΓ

But, in question it is not given that ωΓ<<1. So, is it correct to do this approximation?

B)

In many questions, Irodov asks to find out the angle between velocity and acceleration.
Does this angle have any physical significance?
I mean if I know this angle what can I tell about the motion?
Why is one supposed to know this angle?

Last edited: Jul 19, 2017
2. Jul 19, 2017

### haruspex

Your equation for distance traversed is wrong. It should read ds2=dx2+dy2, not s2=x2+y2.

3. Jul 19, 2017

### Pushoam

Thanks.

Learning , I should be careful while calculating magnitude of the displacement or distance. There is a tendency to get confused between the two.

O.K. So, s is the magnitude of the displacement traveled in time Γ.

ds2 = dx2+dy2

dx = aω cos(ωt) dt

dy = aω sin(ωt) dt

dx2 = (dx) (dx) = [ aω cos(ωt)]2 (dt)2
dy2 = (dy) (dy) = [ aω sin(ωt)]2 (dt)2

ds2 = [ aω ]2 (dt)2

ds = aω dt

0s ds = aω ∫0Γ dt

s = aωΓ
And for ωΓ <<1,the distance is approximately equal to the displacement.

Is this solution correct?

4. Jul 19, 2017

### haruspex

Yes. (But the original uses $\tau$, Greek lowercase tau, which you are writing as $\Gamma$, Greek uppercase gamma.)

5. Jul 19, 2017

### Pushoam

Thanks for this, too, as earlier I thought both Γ and τ are tau's.

6. Jul 19, 2017

### haruspex

Uppercase tau is indistinguishable from T.

7. Jul 19, 2017