Question

A particle moves in the plane so that its velocity and position vectors are always orthogonal. Show that the particle moves in a circle centered at the origin.

Answer 1

Let the velocity vector be v and the position vector be r.

By the definition of dot product, v∙r = 0 = |v|*|r|*cos(θ), since v and r are orthogonal

Also, v = dr/dt
==> (dr/dt)∙r = 0
==> r dr = 0 dt

==> ∫ r dr = ∫ 0 dt
==> r2/2 = C
==> r2 = 2C
here 2C, a constant is equal to R2, another constant.

Then, r2 = R2
Since r is a vector, let's say (x, y), then we have x2 + y2 = R2

This is the equation of a circle centered at the origin.

Hence the position vector describes a circle centered at the origin and the particle must move in a circle centered at the origin.