In this problem, we are asked to find the relative speed between you and the thief. This speed is the distance between you and the thief divided by time, so we need to find the distance between you and the thief, which itself depends on time. Since this
is a 2-dimensional problem, set up an x-y-coordinate system with you initially at the origin. So your initial position is r1=(0,0), the thief's initial position is r2=(8,6), and your distance is Δr=√((8-0)²+(6-0)²)=10 (all numbers are
in meters). The thief moves in the x-direction at 7 m/s, so his position as a function of time will be r2=(8+7t,6).
(a) You move in the positive x-direction at 9 m/s. Your position at time t is r1=(9t,0). Your relative distance is
Δr=√((8+7t-9t)²+(6-0)²)=√((8-2t)²+(6)²) = √(100-32t+4t²)
Notice that this distance is smallest at t=4s, when Δr=6m.
Your relative speed is v=Δr/t =√(100-32t+4t²) /t.
(c) I will do part (c) next, because it's in the same spirit as part (a).
You move in the negative x-direction at 9 m/s. Your position at time t is r1=(-9t,0). Your relative distance is Δr=√((8+7t+9t)²+(6-0)²)=√((8+16t)²+(6)²) = √(100+256t+256t²)
Notice that this distance is smallest at t=0, when Δr=10m.
Your relative speed is v=Δr/t=√(100+256t+256t²) /t.
(b) "Running toward the thief" is a much harder problem, as you no longer follow a straight line path. Clearly, you are crossing the street, but never reach the other side of it. The tangent of the curve along which you are running is always pointing toward
the thief. Such a curve is called a "tractrix" or "dog curve" (as in a dog running towards his owner, who moves along a straight line), and is quite complicated mathematically.
I will leave this part out, unless you really want to see it.