Kinematics - Given the Velocity Function

The velocity \(v\) m s\(^{-1}\) of a particle at time \(t\) seconds, is given by \(v=2t+\cos{2t}\), for \(0\leq t \leq 2\).

1. Write down the velocity of the particle when \(t=0\).[1]

When \(t=k\), the acceleration is zero.

2. 1. Show that \(k=\frac{\pi}{4}\).
   2. Find the exact velocity when \(t=\frac{\pi}{4}\).[8]
3. When \(t<\frac{\pi}{4}\), \(\frac{\mathrm{d}v}{\mathrm{d}t}>0\) and when \(t>\frac{\pi}{4}\), \(\frac{\mathrm{d}v}{\mathrm{d}t}>0\).
   Sketch a graph of \({v}\) against \(t\).[4]
4. Let \(d\) be the distance travelled by the particle for \(0 \leq t \leq 1\).
   1. Write down an expression for \(d\).
   2. Represent \(d\) on your sketch.[3]