2025 Nov TZ3 P2 Q9

A particle \(P\) moves in a straight line so that its displacement, \(s\) cm, from a fixed point \(O\) at time \(t\) seconds is given by \(s(t) = 2^{\left(1 - \frac{t}{5} \right)} \sin{\left(\frac{2\pi t}{3} \right)}\), where \(t \geq 0\).
The following diagram shows part of the graph of \(y = s(t)\).

1. Find

   1. the maximum displacement of \(P\) from \(O\) ;

   2. the maximum velocity of \(P\).

      [5]

2. Find

   1. the minimum value of the displacement function \(s(t)\) ;

   2. the displacement of \(P\) from \(O\) when \(t = 3.5\).

      [3]

3. Hence, determine the total distance travelled by \(P\) in the first \(3.5\) seconds.

   [3]

The first time that \(P\) returns and passes through \(O\) is when \(t = T\).

4. Write down the value of \(T\).

   [1]

The particle passes through \(O\) every \(T\) seconds.
A sequence \(u_1, u_2, u_3 \dots\) is formed where \(u_1, u_2, u_3 \dots\) are the largest distances from \(O\) in each of the intervals \(0 < t < T, T < t < 2T, 2T < t < 3T \dots\) respectively.
It is known that \(u_1, u_2, u_3 \dots\) form a geometric sequence.

5. 1. Determine the value of the common ratio \(r\) of this geometric sequence.

   2. Calculate the total distance travelled by the particle if it were to continue to move in this way indefinitely.

      [5]