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The problem presented describes a particle moving along a straight line with the position-time relationship \( x(t) = -\frac{1}{2}t^2 + 23t - 204 \), where time is in seconds and position is in meters. The following list of questions pertains to this motion, and I will address each one step by step.

### 3.1 Initial Position
The initial position is determined by evaluating \( x(t) \) at \( t = 0 \).
\[
x(0) = -\frac{1}{2}(0)^2 + 23(0) - 204 = -204 \text{ meters}
\]
Thus, the initial position is \(-204\) meters.

### 3.2 When Does the Particle Pass the Origin?
To find when the particle passes the origin, set \( x(t) = 0 \) and solve for \( t \).
\[
0 = -\frac{1}{2}t^2 + 23t - 204
\]
Multiply the equation by 2 to simplify:
\[
0 = -t^2 + 46t - 408
\]
Use the quadratic formula:
\[
t = \frac{-46 \pm \sqrt{46^2 - 4(-1)(-408)}}{2(-1)} = \frac{-46 \pm \sqrt{2116 - 1632}}{-2} = \frac{-46 \pm \sqrt{484}}{-2} = \frac{-46 \pm 22}{-2}
\]
The two solutions are:
\[
t = \frac{-46 + 22}{-2} = 12 \quad \text{and} \quad t = \frac{-46 - 22}{-2} = 34
\]
Therefore, the particle passes the origin at \( t = 12 \) seconds and \( t = 34 \) seconds.

### 3.3 When Does the Particle Exit?
By "exit," I assume the problem refers to the second time the particle crosses the origin, which is at \( t = 34 \) seconds, as calculated above.

### 3.4 When Does the Particle Change Direction?
The particle changes direction when its velocity equals zero. Velocity is the derivative of position:
\[
v(t) = \frac{dx}{dt} = -t + 23
\]
Set \( v(t) = 0 \) to find when the direction changes:
\[
0 = -t + 23 \quad \Rightarrow \quad t = 23 \text{ seconds}
\]
Thus, the particle changes direction at \( t = 23 \) seconds.

### 3.5 Maximum Positive Displacement
The maximum displacement occurs at \( t = 23 \) seconds (the turning point). Substituting \( t = 23 \) into the position equation:
\[
x(23) = -\frac{1}{2}(23)^2 + 23(23) - 204 = -\frac{1}{2}(529) + 529 - 204 = -264.5 + 529 - 204 = 60.5 \text{ meters}
\]
Thus, the maximum positive displacement is 60.5 meters.

Would you like to proceed with the rest of the questions, or do you have any specific queries about this so far?

Given a particle moving along a straight line, its position-time equation is given by x(t) = -1/2 t^2 + 23t - 204, where time is in seconds and position is in meters. Several questions about the motion are presented, including when the particle passes the origin, its maximum displacement, and velocity relationships.

Learn how to solve a quadratic position-time equation describing a particle's motion, determining key points such as initial position, time to pass the origin, maximum displacement, and velocity relationships. Suitable for high school students.

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