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# Compute the velocity and acceleration of a particle

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.

1. The velocity of the particle at time .
2. The acceleration of the particle at time .
3. The acceleration of the particle at time .
4. The displacement of the particle from its initial position 0 at time .
5. Find a time such that the particle returns to its initial position, or prove that no such time exists.

1. Since the velocity of the particle is given by the derivative of the position we can compute,

2. Next, the acceleration is the derivative of the velocity so using part (a) we have

3. We know from the problem statement that for where is a constant. Hence, for . Thus, the acceleration at time is 0.
4. 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

5. Finally, the position of the particle at time is given by

Setting this equal to 0 and solving for time,