Let's tackle each part of this problem step by step.

Part (a): Initial Displacement

The initial displacement occurs at $t = 0$. Using the function for $s(t)$: $s(0) = 3(0 + 2)^{\cos(0)}$ Since $\cos(0) = 1$: $s(0) = 3 \times 2^1 = 6 \, \text{meters}$ So, the initial displacement is 6 meters.

Part (b): Velocity at $t = 2$

The velocity is the derivative of the displacement function $s(t)$. Let’s differentiate $s(t) = 3(t + 2)^{\cos t}$ with respect to $t$ using the chain rule and logarithmic differentiation. After finding $s'(t)$, substitute $t = 2$ to find the velocity.

Part (c): Intervals of Motion Away from Point P

To find when the particle is moving away from $P$, check when $s'(t) > 0$. This involves analyzing the derivative $s'(t)$ over the interval $0 \leq t \leq 6.8$ and identifying where it is positive.

Would you like a detailed breakdown of the differentiation process and interval analysis?

Further Questions

1. How do you determine the direction of particle motion from velocity?
2. Why is logarithmic differentiation helpful here?
3. What methods can be used to analyze the intervals of motion?
4. What role does the cosine function play in determining behavior over time?
5. How would the initial displacement change if the function's base was altered?

Tip: Use derivative sign analysis to understand motion direction.