Further Mechanics 2: Further dynamicsEdexcel A-Level Further Maths: Topic test

1. 1
   A raft of mass 150 kg is pulled across still water by a constant horizontal force of 600 N. The water exerts a resistance of $30v$ N, where $v$ m s⁻¹ is the speed of the raft. The raft starts from rest at time $t=0$, where $t$ is in seconds.

   (a)

   What is the limiting speed of the raft?

   [1 mark]

   • A$20$ m s⁻¹
   • B$4$ m s⁻¹
   • C$5$ m s⁻¹
   • D$0.2$ m s⁻¹

   (b)

   Which expression gives $v$ at time $t$?

   [1 mark]

   • A$20\mathrm e^{-0.2t}$
   • B$4\left(1-\mathrm e^{-5t}\right)$
   • C$20\left(1-\mathrm e^{-0.2t}\right)$
   • D$20\left(1-\mathrm e^{-5t}\right)$

   (c)

   Find the time taken for the raft to reach a speed of 10 m s⁻¹.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A buoy rises and falls vertically with simple harmonic motion. It moves between a highest point 0.6 m above its mean level and a lowest point 0.6 m below it, and one complete oscillation takes 4 seconds.

   (a)

   What is the angular frequency of the motion?

   [1 mark]

   • A$\frac{2}{\pi}$ rad s⁻¹
   • B$\frac{\pi}{2}$ rad s⁻¹
   • C$8\pi$ rad s⁻¹
   • D$\frac{\pi}{4}$ rad s⁻¹

   (b)

   What is the maximum speed of the buoy?

   [1 mark]

   • A$1.48$ m s⁻¹
   • B$0.382$ m s⁻¹
   • C$1.88$ m s⁻¹
   • D$0.942$ m s⁻¹

   (c)

   Find the speed of the buoy when it is 0.3 m above its mean level.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass 0.5 kg moves along the positive $x$-axis. When $P$ is at the point with coordinate $x$ metres, the only force acting on $P$ is a repulsive force of magnitude $\frac{8}{x^2}$ newtons directed away from the origin $O$. At the point where $x=1$, $P$ has speed 2 m s⁻¹ and is moving away from $O$.

   (a)

   Show that $v^2=36-\frac{32}{x}$, where $v$ m s⁻¹ is the speed of $P$ at the point with coordinate $x$.

   [3 marks]

   (b)

   Show that the speed of $P$ never reaches 6 m s⁻¹, and find the speed of $P$ when $x=4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A uniform cylindrical float of mass 0.4 kg and cross-sectional area 0.002 m² floats upright in water of density 1000 kg m⁻³. The float is displaced vertically and released, and it remains partly submerged throughout its motion. The upthrust on the float is equal to the weight of the water it displaces. Take $g=9.8$ m s⁻².

   (a)

   The float is displaced a distance $x$ metres downwards from its equilibrium position. Show that the float moves with simple harmonic motion and find the period of the motion.

   [6 marks]

   (b)

   The float is released from rest at a point 0.05 m below its equilibrium position. Find the maximum speed of the float and the time taken for it to first reach a point 0.025 m below its equilibrium position.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A particle of mass 4 kg is at rest at the origin $O$ on a smooth horizontal surface. For $t\ge0$, where $t$ is in seconds, a horizontal force of magnitude $8\mathrm e^{-2t}$ newtons acts on the particle in a fixed direction.

   (a)

   What is the initial acceleration of the particle?

   [1 mark]

   • A$8$ m s⁻²
   • B$32$ m s⁻²
   • C$0.5$ m s⁻²
   • D$2$ m s⁻²

   (b)

   Which expression gives the velocity of the particle at time $t$?

   [1 mark]

   • A$2\mathrm e^{-2t}$
   • B$1-\mathrm e^{-2t}$
   • C$4\left(1-\mathrm e^{-2t}\right)$
   • D$-\mathrm e^{-2t}$

   (c)

   Find the distance of the particle from $O$ when $t=2$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The piston of a small pump moves in a straight line so that its displacement, in metres, from the centre of its stroke at time $t$ seconds is $x=0.5\sin2t$.

   (a)

   What is the maximum acceleration of the piston?

   [1 mark]

   • A$1$ m s⁻²
   • B$4$ m s⁻²
   • C$2$ m s⁻²
   • D$0.5$ m s⁻²

   (b)

   What is the first time, for $t>0$, at which the displacement is 0.25 m?

   [1 mark]

   • A$0.262$ s
   • B$0.524$ s
   • C$0.250$ s
   • D$1.31$ s

   (c)

   Find the speed of the piston when its displacement is 0.3 m.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A probe of mass $m$ is released from rest at a distance $2R$ from the centre of Mars and moves directly towards the centre. Mars is modelled as a fixed sphere of radius $R=3.4\times10^6$ m, and the gravitational acceleration at its surface is $g=3.7$ m s⁻². The only force on the probe is the gravitational attraction of Mars, which is inversely proportional to the square of the distance $x$ of the probe from the centre of Mars.

   (a)

   Show that, at distance $x$ from the centre of Mars, the speed $v$ of the probe satisfies $v^2=\frac{2gR^2}{x}-gR$.

   [3 marks]

   (b)

   Find the speed with which the probe reaches the surface of Mars. Find also the speed with which it would reach the surface if it were released from rest at a distance $4R$ from the centre.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A uniform spherical asteroid has radius $R=5.0\times10^5$ m, and the gravitational acceleration at its surface is $g=0.80$ m s⁻². A smooth straight tunnel is drilled through the asteroid along a diameter, and a probe of mass $m$ is released from rest at one end of the tunnel. When the probe is at a distance $x$ from the centre of the asteroid, with $x\le R$, the gravitational force on it has magnitude $\frac{mgx}{R}$ and is directed towards the centre.

   (a)

   Show that the probe moves with simple harmonic motion and find the period of the motion.

   [6 marks]

   (b)

   Find the maximum speed of the probe and the time taken for it to travel from the end of the tunnel to the point halfway between that end and the centre of the asteroid.

   [6 marks]

   Total for question 8: 12 marks

End of questions