2025 May TZ1 P1 Q7

An object moves in a straight line.
Its velocity \(\mathrm{v}\) m s\(^{-1}\), at time \(t\) seconds, is given by \(\mathrm{v}(t) = 30 + 20t - 10t^2\) for \(0 \leq t \leq 5\).
The graph of \(\mathrm{v}\) is shown in the following diagram.

The graph of \(\mathrm{v}\) has a local maximum point where \(t = 1\) and intersects the \(t\)-axis at \(t = 3\).

1. Determine the object's

   1. maximum velocity;

   2. maximum speed.

      [4]

At \(t = T\), the object changes direction.

2. 1. Write down the value of \(T\).

   2. Find the distance travelled by the object in the first \(T\) seconds.

      [5]

3. Determine whether the object returns to its initial position during the time period \(0 \leq t \leq 5\), justifying your answer.

   [4]