Calculating distance, angle bet. velocity and acceleration

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

A) [/B]

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 = aωΓBut, 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?

 

[haruspex]

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

 

[Pushoam]

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

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

∫₀^s ds = aω ∫₀^Γ dt

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

Is this solution correct?

 

[haruspex]

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

∫₀^s ds = aω ∫₀^Γ dt

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

Is this solution correct?

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

 

[Pushoam]

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

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

 

[haruspex]

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

Uppercase tau is indistinguishable from T.

 

[Pushoam]

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?

What about this question?

 

[haruspex]

What about this question?

Not sure that the angle has any general meaning in itself. Certainly you can say interesting things about the extreme cases (collinear and orthogonal), and likewise regarding the sine and cosine of the angle.