Centripetal acceleration geometry

[richardbsmith]

This is probably a geometry question more that a physics question. I am trying to prove that in uniform circular motion \Delta V/V= s/R.

I am basically trying to show that S forms a right triangle with \DeltaV, when V{1} is added to V{2} as a vector. (This is to demonstrate that the triangles are similar.)

I understand that the angle formed by S and \DeltaV is a right angle because it obviously inscribes the diameter. I just cannot seem to find a satisfactory proof that \DeltaV must necessarily intersect the circle at the diameter.

Probably not explaining this very well.

 

[Lojzek]

This is probably a geometry question more that a physics question. I am trying to prove that in uniform circular motion \Delta V/V= s/R.

I am basically trying to show that S forms a right triangle with \DeltaV, when V{1} is added to V{2} as a vector. (This is to demonstrate that the triangles are similar.)

I understand that the angle formed by S and \DeltaV is a right angle because it obviously inscribes the diameter. I just cannot seem to find a satisfactory proof that \DeltaV must necessarily intersect the circle at the diameter.

Probably not explaining this very well.

If you are solving geometry problems with both distances and velocities involved, then you
are probably making a mistake: remember that they have different units! (unless you study relativistic theory)
For the mentioned problem you should use formulas:
s=R*fi (fi is angle of the part of orbit traveled in radians)
\DeltaV=V*sin(fi') (fi' is the angle between the old and new velocity vector)

Prove that fi=fi' and use sin(fi)=fi (for small angles) and you will get \Delta V/V= s/R

 

[richardbsmith]

Thank you so much for responding. I think though my question which started with uniform motion and delta V, is now simply a geometry question.

I will try to put up a drawing of what I so pitifully tried to explain.

Here is another image with a different angle and size of the tangents.

I have tried several approaches, but I cannot prove that the angle formed from tangent 1 to tangent 2 to the circle must inscribe a 180 degree arc and must be a right angle.

 

[A.T.]

From : http://en.wikipedia.org/wiki/Inscribed_angle_theorem

In geometry, the inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.

So, to get an inscribed angle of 90° you need a central angle of 180°(=diameter line).