Kinematics using calculusEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle $P$ moves along a straight line. Its displacement $x$ metres from a fixed point $O$ at time $t$ seconds is given by $x=t^3-6t^2+9t$, for $t\geq0$.

   (a)

   At which times is $P$ instantaneously at rest?

   [1 mark]

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

   (b)

   What is the acceleration of $P$ when $t=3$?

   [1 mark]

   • A$0\text{ m s}^{-2}$
   • B$18\text{ m s}^{-2}$
   • C$-6\text{ m s}^{-2}$
   • D$6\text{ m s}^{-2}$

   (c)

   Find the displacement of $P$ from $O$ at the instant when its acceleration is zero.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $Q$ moves along a straight line through a fixed point $O$. At time $t$ seconds its velocity is $v=6t-3t^2$ in $\text{m s}^{-1}$, for $0\leq t\leq3$, and $Q$ is at $O$ when $t=0$.

   (a)

   What is the displacement of $Q$ from $O$ when $t=2$?

   [1 mark]

   • A$12$ m
   • B$8$ m
   • C$4$ m
   • D$6$ m

   (b)

   What is the acceleration of $Q$ when $t=2$?

   [1 mark]

   • A$-6\text{ m s}^{-2}$
   • B$0\text{ m s}^{-2}$
   • C$6\text{ m s}^{-2}$
   • D$-12\text{ m s}^{-2}$

   (c)

   Find the greatest velocity of $Q$ for $0\leq t\leq3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves in a horizontal plane. At time $t$ seconds its position vector relative to a fixed origin $O$ is $\mathbf{r}=(2t^3-3t)\mathbf{i}+(t^2+4t)\mathbf{j}$ metres, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

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

   [3 marks]

   (b)

   Find the speed of $P$ when $t=2$, and the magnitude of the acceleration of $P$ when $t=1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ moves along a straight line. At time $t=0$ it passes through the point $O$ with velocity $24\text{ m s}^{-1}$. At time $t$ seconds its acceleration is $(6t-18)\text{ m s}^{-2}$ in the direction of increasing displacement $x$ from $O$.

   (a)

   (i) Show that $v=3t^2-18t+24$.
   (ii) Find the times when $P$ is instantaneously at rest.
   (iii) Find the displacement of $P$ from $O$ at the later of these times.

   [6 marks]

   (b)

   Find the total distance travelled by $P$ in the first $5$ seconds of its motion.

   [6 marks]

   Total for question 4: 12 marks

End of questions