(a) The first particle shoots off to -infinity ar t=3 (since this is the singular
point in the equation for its position). This is the time required for this part.
Note with this equation for position at t=3 the second particle flies in from
+ infinity, and this does not make much sense to me.
(b) The first particle leaves the chamber when s(t) = 0 for the second time,
so we need to solve:
4t + 2/(t-3) + (2/3) = 0
multiply through by (t-3) to get:
4t^2 -12t +2 + 2/3 t -2 = 0
4 t^2 -34/3 t =0
which has roots t=0, and t=17/6 ns.
So it leave the chamber at 16/6 ns.
(c) The change directions when ds/dt=0.
ds/dt = 4 + 2 (-1)/(t-3)^2
so they change direction at the roots of:
4 - 2/(t-3)^2 = 0.
Multiply through by (t-3)^2 (we can do this as we know the roots we seek
do not occur at t=3) to get:
4(t-3)^2 - 2 = 0
2 t^2 - 12 t + 17 =0
which from the quadratic formula has roots 3-1/sqrt(2) and 3+1/sqrt(2).
Hence the particles change directions at 3-1/sqrt(2) and 3+1/sqrt(2) ns.
(d) The composite curve is a hyperbola.