Calculus in kinematicsEdexcel A-Level Maths: Subtopic test

1. 1
   A particle $P$ moves along a straight line. At time $t$ seconds, $t\ge0$, its displacement from a fixed point $O$ is $s=t^3-6t^2+9t$ metres.

   (a)

   What is the velocity of $P$ when $t=2$?

   [1 mark]

   • A$-3$ m s$^{-1}$
   • B$2$ m s$^{-1}$
   • C$9$ m s$^{-1}$
   • D$12$ m s$^{-1}$

   (b)

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

   [1 mark]

   • A$-12$ m s$^{-2}$
   • B$0$ m s$^{-2}$
   • C$-3$ m s$^{-2}$
   • D$12$ m s$^{-2}$

   (c)

   Find the times at which $P$ is instantaneously at rest.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves along a straight line through a point $O$. At time $t$ seconds, $t\ge0$, its velocity is $v=6t-3t^2$ m s$^{-1}$, and at $t=0$ it is at $O$.

   (a)

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

   [1 mark]

   • A$0$ m
   • B$8$ m
   • C$4$ m
   • D$20$ m

   (b)

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

   [1 mark]

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

   (c)

   Find the time, after $t=0$, at which the particle returns to $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves along a straight line. At time $t$ seconds, $t\ge0$, its displacement from a fixed point $O$ is $s=2t^3-9t^2+12t$ metres.

   (a)

   Find expressions for the velocity and the acceleration of $P$ at time $t$.

   [3 marks]

   (b)

   Find the times at which $P$ is instantaneously at rest, and its displacement from $O$ at each of these times.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves in a straight line. At $t=0$ it is at the point $O$ with velocity $2$ m s$^{-1}$. Its acceleration at time $t$ seconds is $a=6t-4$ m s$^{-2}$, for $t\ge0$.

   (a)

   (i) Find an expression for the velocity of the particle at time $t$.
   (ii) Find an expression for the displacement of the particle from $O$ at time $t$.
   (iii) Find the displacement from $O$ when $t=3$.

   [6 marks]

   (b)

   (i) Show that the particle never changes its direction of motion.
   (ii) Find the velocity of the particle at the instant when its acceleration is zero.
   (iii) Hence state the distance travelled in the first 3 s, giving a reason.

   [6 marks]

   Total for question 4: 12 marks

End of questions