Mechanics 3 (M3): Further kinematicsEdexcel International A Level Further Maths: Topic test

1. 1
   A particle $P$ moves along the $x$-axis. At time $t$ seconds, $t\ge0$, the acceleration of $P$ is $(6t-4)$ m s$^{-2}$ in the positive $x$ direction. When $t=0$, $P$ is at the origin $O$ and has velocity $3$ m s$^{-1}$ in the positive $x$ direction.

   (a)

   Find the velocity of $P$ when $t=2$, in m s$^{-1}$.

   [1 mark]

   • A$4$
   • B$8$
   • C$7$
   • D$19$

   (b)

   Find the displacement of $P$ from $O$ when $t=2$, in m.

   [1 mark]

   • A$6$
   • B$7$
   • C$10$
   • D$0$

   (c)

   Find the least speed of $P$ for $t\ge0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ moves along the $x$-axis. When $P$ is at displacement $x$ metres from the origin $O$, its acceleration is $-4x$ m s$^{-2}$ in the positive $x$ direction. When $P$ is at $O$, its speed is $8$ m s$^{-1}$ and it is moving in the positive $x$ direction.

   (a)

   Find the speed of $P$ when $x=2$, in m s$^{-1}$ to 3 significant figures.

   [1 mark]

   • A$7.48$
   • B$6.93$
   • C$0$
   • D$8$

   (b)

   Find the greatest distance of $P$ from $O$ in the subsequent motion, in m.

   [1 mark]

   • A$8$
   • B$16$
   • C$2$
   • D$4$

   (c)

   Find the magnitude of the acceleration of $P$ at an instant when its speed is $6$ m s$^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves along the $x$-axis. At time $t$ seconds, $t\ge0$, the velocity of $P$ is $(3t^2-12t+9)$ m s$^{-1}$ in the positive $x$ direction. When $t=0$, $P$ is at the point where $x=2$.

   (a)

   Find an expression for the displacement $x$ metres of $P$ from $O$ at time $t$ seconds.

   [3 marks]

   (b)

   Find the total distance travelled by $P$ in the first $4$ seconds.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ moves along the $x$-axis. When $P$ is at displacement $x$ metres from the origin $O$, its acceleration is $(6-2x)$ m s$^{-2}$ in the positive $x$ direction. When $P$ is at $O$, its speed is $4$ m s$^{-1}$ and it is moving in the positive $x$ direction.

   (a)

   (i) Show that $v^2=16+12x-2x^2$, where $v$ m s$^{-1}$ is the velocity of $P$.
   (ii) Find the greatest speed of $P$ in the subsequent motion.

   [6 marks]

   (b)

   (i) Find the greatest distance of $P$ from $O$ in the positive $x$ direction.
   (ii) Find the acceleration of $P$ at this point, and the speed of $P$ when it next passes through $O$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A particle $P$ moves along the $x$-axis. At time $t$ seconds, $t\ge0$, the acceleration of $P$ is $8\mathrm{e}^{-2t}$ m s$^{-2}$ in the positive $x$ direction. When $t=0$, $P$ is at the origin $O$ and is at rest.

   (a)

   Find the velocity of $P$ when $t=\ln2$, in m s$^{-1}$.

   [1 mark]

   • A$5$
   • B$3$
   • C$-1$
   • D$2$

   (b)

   Find the displacement of $P$ from $O$ when $t=\ln2$, in m to 3 significant figures.

   [1 mark]

   • A$3.27$
   • B$4.27$
   • C$1.27$
   • D$0.773$

   (c)

   Find the time at which the acceleration of $P$ is $1$ m s$^{-2}$, in seconds to 3 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A particle $P$ moves along the positive $x$-axis. When $P$ is at distance $x$ metres from the origin $O$, its acceleration is $\frac{2}{x}$ m s$^{-2}$ in the positive $x$ direction. When $x=1$, $P$ has speed $3$ m s$^{-1}$ and is moving in the positive $x$ direction.

   (a)

   Find the speed of $P$ when $x=\mathrm{e}^2$, in m s$^{-1}$ to 3 significant figures.

   [1 mark]

   • A$3.61$
   • B$7$
   • C$17$
   • D$4.12$

   (b)

   Find the value of $x$ at which the speed of $P$ is $5$ m s$^{-1}$, to 3 significant figures.

   [1 mark]

   • A$54.6$
   • B$7.39$
   • C$2980$
   • D$4$

   (c)

   Find the greatest value of $x$ for which the speed of $P$ is less than $6$ m s$^{-1}$, to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ moves along the positive $x$-axis. At time $t$ seconds, $t\ge0$, the displacement of $P$ from the origin $O$ is $x$ metres, where $\frac{dx}{dt}=\frac{6}{x+1}$. When $t=0$, $P$ is at $O$.

   (a)

   Show that $x^2+2x=12t$.

   [3 marks]

   (b)

   Find the time at which $P$ is $4$ m from $O$, and the velocity of $P$ at that time.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ moves along the $x$-axis. At time $t$ seconds, $t\ge0$, the acceleration of $P$ is $6\sin2t$ m s$^{-2}$ in the positive $x$ direction. When $t=0$, $P$ is at the point where $x=1$ and has velocity $-3$ m s$^{-1}$.

   (a)

   (i) Find the velocity of $P$ at time $t$ seconds.
   (ii) Find the first time at which $P$ is instantaneously at rest, and the acceleration of $P$ at that time.

   [6 marks]

   (b)

   (i) Find an expression for the displacement $x$ metres of $P$ from $O$ at time $t$ seconds.
   (ii) Find the first time $t>0$ at which $P$ passes through $O$, and the velocity of $P$ at that time.

   [6 marks]

   Total for question 8: 12 marks

End of questions