A mechanism moves a particle along a straight line with displacement (from an initial position 0) at time given by the function
At time the propelling mechanism stops working, and the particle then moves with constant velocity (the velocity imparted by the mechanism at time
). Compute the following.
- The velocity of the particle at time
.
- The acceleration of the particle at time
.
- The acceleration of the particle at time
.
- The displacement of the particle from its initial position 0 at time
.
- Find a time
such that the particle returns to its initial position, or prove that no such time exists.
- Since the velocity of the particle is given by the derivative of the position we can compute,
- Next, the acceleration is the derivative of the velocity so using part (a) we have
- We know from the problem statement that
for
where
is a constant. Hence,
for
. Thus, the acceleration at time
is 0.
- To find the position at time
we consider the motion of the particle over two time intervals: the time from 0 to
and the time from
to
. During the time from 0 to
the position is given by the function
From time
we know the particle then moves with constant velocity of
so its position changes by
during the time interval from
to
. Therefore, the position at time
is given by
- Finally, the position of the particle at time
is given by
Setting this equal to 0 and solving for time,