tmua.me Extreme Set C Paper 2 · Question 18

A particle moves in a plane. At time $t\geq0$ , its coordinates are $x(t)$ and $y(t)$ , where

x(0)=y(0)=0,\qquad \frac{\mathrm dx}{\mathrm dt}=1-t^2,\qquad \frac{\mathrm dy}{\mathrm dt}=2t.

Its speed is

\sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+ \left(\frac{\mathrm dy}{\mathrm dt}\right)^2}.

For $T>0$ , let $D(T)$ be the total distance travelled during $0\leq t\leq T$ , and let $R(T)$ be the straight-line distance of the particle from the origin at time $T$ .

What is the greatest possible value of $D(T)/R(T)$ ?

\dfrac2{\sqrt3}

1

\dfrac4{\sqrt{13}}

\dfrac43

\sqrt2

examday

tmua.me Extreme Set C Paper 2 · Q18