At time
, the particle has position
, so
You also know that at this point, when
, the particle's speed is
, which is the magnitude of the velocity at this point because the particle is traveling in one direction. This means
If
is the time at which the particle arrives at the point
, i.e.
, then you have the system
Eliminating the terms with
, you're left with the system
so you're left with solving this system with the constraint that the sum of these constants' squares is
.
However, there are two solutions to this, with
Obviously, we require that
, so we need to check if either option forces using an invalid value of
. This amounts to finding
such that
. Indeed, this has two solutions,
, or
and
. We ignore the first.
Plugging in this value of
into the system
, we find that
Therefore the equation for the position vector is