This question is in the book Analytical Mechanics (7th) chp 1.27 I don\'t follow

Question

This question is in the book Analytical Mechanics (7th) chp 1.27
I don't follow/understand the solution given by cramster. Can someone please elaborate on it, or provide a better one. I'm not trying to copy and paste this, I really want to understand it.
Thanks.

|V x a| = v^3/p

I'm pretty sure V is the vector for velocity, and v is instantanious velocity. p is the radius of curvature of the path of a moving particle.

*Note. To clarify, I understand everything up to the very end. where I have:
|V x a| = (v^3/p)*sin(theta)
I konw the proof is true, so why is the next step assuming that sin(theta)=1.
What are we assuming the angle between V and a is? This question is in the book Analytical Mechanics (7th) chp 1.27
I don't follow/understand the solution given by cramster. Can someone please elaborate on it, or provide a better one. I'm not trying to copy and paste this, I really want to understand it.
Thanks. |V x a| = v^3/p I'm pretty sure V is the vector for velocity, and v is instantanious velocity. p is the radius of curvature of the path of a moving particle. *Note. To clarify, I understand everything up to the very end. where I have: |V x a| = (v^3/p)*sin(theta) I konw the proof is true, so why is the next step assuming that sin(theta)=1. What are we assuming the angle between V and a is?

Explanation / Answer

The a is acceleration as defined is normal to the velocity, so its always 90 degrees, making the sin always 1. To say it in another way, the Centripetal acceleration is in the radial direction and the velocity as defined is in the tangential direction, the two directions by definition are always at a right angle.

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