Question

A man runs round a circular track at a constant speed.
Would like to know how to drawn locus of the points representing the magnitude ‘d’ of the displacement of the man from the starting position against time ‘t’ , during one lap of the track.

Answer 1

consider the angle θ subtended by the arc he has run on a track of radius r. The location of the point at time t is s*t where s is his speed in units of distance/time.

If we lay out the track so he started at (r,0) and ran in a counterclockwise direction, then at time t, his location is

(x,y) = (r cosθ,r sinθ)
and θ st/r

the distance from (r,0) is thus

d = √((r-x)^2 + y^2)
= r√((1-cosθ)^2 + sin^2θ)
= r√(2-2cosθ)
= 2r sin(θ/2)
= 2r sin(st/2r))