1. 1
   A particle moves along a straight line. Its displacement from a fixed point O after $t$ seconds is $s(t)=t^{3}-6t^{2}+9t$ metres, for $t\ge0$.

   (a)

   Find the velocity of the particle when $t=2$.

   [1 mark]

   • A$2$ m s$^{-1}$
   • B$0$ m s$^{-1}$
   • C$3$ m s$^{-1}$
   • D$-3$ m s$^{-1}$

   (b)

   Find the times at which the particle is instantaneously at rest.

   [1 mark]

   • A$t=1$ and $t=3$
   • B$t=2$ only
   • C$t=0$ and $t=3$
   • D$t=-1$ and $t=-3$

   (c)

   Find the acceleration of the particle when it is first at rest.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cyclist rides along a straight road. Her velocity $t$ seconds after passing a marker is $v(t)=8-2t$ m s$^{-1}$, for $0\le t\le6$. Displacement is measured from the marker, positive in the direction she was travelling when she passed it.

   (a)

   Find the acceleration of the cyclist when $t=3$.

   [1 mark]

   • A$2$ m s$^{-2}$
   • B$-2$ m s$^{-2}$
   • C$8$ m s$^{-2}$
   • D$-6$ m s$^{-2}$

   (b)

   Find the displacement of the cyclist from the marker when $t=6$.

   [1 mark]

   • A$20$ m
   • B$16$ m
   • C$12$ m
   • D$-4$ m

   (c)

   Find the total distance travelled by the cyclist for $0\le t\le6$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle moves in a straight line with acceleration $a(t)=6t-4$ m s$^{-2}$, for $t\ge0$. When $t=0$, its velocity is $1$ m s$^{-1}$ and its displacement from a fixed point O is $2$ m.

   (a)

   Find an expression for the velocity $v(t)$.

   [3 marks]

   (b)

   The particle is at rest when $t=\frac{1}{3}$ and when $t=1$.
   (i) Find the displacement of the particle from O when $t=1$.
   (ii) Describe the direction of motion of the particle for $\frac{1}{3}<t<1$, justifying your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An object hanging on a spring moves up and down along a vertical line. Its displacement above its rest position $t$ seconds after it is released is $s(t)=10\cos\left(\frac{\pi t}{4}\right)$ cm, for $t\ge0$.

   (a)

   (i) Find expressions for the velocity $v(t)$ and the acceleration $a(t)$ of the object.
   (ii) Show that $a(t)=-ks(t)$, where $k$ is a positive constant to be found.
   (iii) Hence explain why the acceleration always acts towards the rest position.

   [6 marks]

   (b)

   (i) Find the first time $t>0$ at which the object is instantaneously at rest, and its displacement at this time.
   (ii) Find the total distance travelled by the object in the first 6 seconds.

   [6 marks]

   Total for question 4: 12 marks

End of questions