Piecewise Motion
In general the motion of a particle is not smooth. Particles may change direction suddenly, collide, a string may become taut or new forces appear, thus changing the acceleration. In these circumstances, we can define for example, the velocity over certain intervals of time.
These points should be noted:
The displacement is always continuous. If the displacement is defined over several time intervals, the ends should 'join' up, so the particle never instantaneously disappears and reappears somewhere else.
Velocity can only change instantaneously in collisions, which may include situations where a string becomes taut, and acceleration can only change instantaneously when forces appear or disappear.
An insect moves in a straight line. The velocity is given by
when
i)Show that
ii)Find
and
and describe the motion for
iii)Sketch
against
for
i)There are no collisions so the velocity is continuous at
Substituting
into
will give the same result as substitutinginto
hence
ii)When
hence
When
hence
(2)-(1) gives
Then from (1)
In the interval
the insect decelerates from 12 m/s to -8m/s.
iii)The graph of v against t is drawn below.
