Perpendicular Velocity and Displacement in Particle Motion

[tifa8]

Hi ! I have a intemediate calculus problem. I am seeing right now all about curves and motion curves.

Homework Statement

Show that 1. the path of a particle lies on a sphere if its displacement and velocity are always perpendicular
2.show that if the particle moves with constant speed then the velocity and acceleration are penperdicular.

Homework Equations

The Attempt at a Solution

1.I think that it is obvious that it lies on a sphere. however I really don't know how to demonstrate that...

if i consider the displacement as r(t)=xi+yj+zk
then r(t).v(t)=0
=>r(t).r'(t)=0
=>xx'+yy'+zz'=0

but it is totally different from a sphere equation which is x^2+y^2+z^2=R^2

2. I don't know if my reasoning is true
since v is constant then v' is equal to zero, thus a is equal to zero

so a.v=0.v=0 so a and v are perpendicular. However, a is a zero vector which I think is quite strange...

thank you for your help

 

[tifa8]

I think I found the answer for question 2.

vector v=Su where u is a unit tangent vector
since the speed S is constant, then S'=0

Or vector a= S'u+K(s)^2N where N is a unit normal vector and K is the curvature
=> a=K(s)^2N

so a.v=K(s)^2N.Su=K(s)^3 N.u=0 since N.u=0 (perpendicular to each other)