Variable acceleration in a straight lineEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle $P$ moves along the $x$-axis. At time $t$ seconds, $t\ge0$, its acceleration is $(6t-12)$ m s$^{-2}$ in the positive $x$-direction. When $t=0$, $P$ is at the origin $O$ and has velocity $9$ m s$^{-1}$.

   (a)

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

   [1 mark]

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

   (b)

   At which times is $P$ instantaneously at rest?

   [1 mark]

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

   (c)

   Find the displacement of $P$ from $O$ when $t=1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ moves along the $x$-axis. When $P$ is at distance $x$ metres from the origin $O$, its acceleration is $(3-x)$ m s$^{-2}$ in the positive $x$-direction. When $t=0$, $P$ is at $O$ moving with speed $4$ m s$^{-1}$ in the positive $x$-direction.

   (a)

   Find the speed of $P$ when $x=3$.

   [1 mark]

   • A$8.5$ m s$^{-1}$
   • B$\sqrt{20.5}$ m s$^{-1}$
   • C$5$ m s$^{-1}$
   • D$4$ m s$^{-1}$

   (b)

   Find the greatest distance of $P$ from $O$.

   [1 mark]

   • A$3$ m
   • B$8$ m
   • C$6$ m
   • D$16$ m

   (c)

   Find the magnitude and direction of the acceleration of $P$ when it is at its greatest distance from $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves along the positive $x$-axis. When $P$ is at distance $x$ metres from the origin $O$, its velocity is $2x$ m s$^{-1}$ in the positive $x$-direction. When $t=0$, $x=3$.

   (a)

   Show that $x=3e^{2t}$.

   [3 marks]

   (b)

   Find the time at which $x=12$, and the acceleration of $P$ at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ moves in a straight line along the $x$-axis, with $t$ in seconds and $x$ in metres.

   (a)

   The acceleration of $P$ at time $t$ is $-8\sin2t$ m s$^{-2}$. When $t=0$, $P$ is at $O$ with velocity $4$ m s$^{-1}$. Find the greatest distance of $P$ from $O$, justifying your answer.

   [6 marks]

   (b)

   In a second motion, $P$ moves along the positive $x$-axis with acceleration $-\frac{16}{x^3}$ m s$^{-2}$ when it is at distance $x$ from $O$. When $t=0$, $x=1$ and $P$ has velocity $4$ m s$^{-1}$ in the positive $x$-direction. Show that $v=\frac4x$, and find the time at which $x=3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions