Variable forces and motion in one dimensionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A particle of mass 2 kg moves in a straight line on a smooth horizontal surface. It starts from rest at the point $O$ and is acted on by a horizontal force of magnitude $6t$ newtons in the direction of motion, where $t$ is the time in seconds after the start.

   (a)

   Find the speed of the particle when $t=2$.

   [1 mark]

   • A$12$ m s⁻¹
   • B$6$ m s⁻¹
   • C$3$ m s⁻¹
   • D$24$ m s⁻¹

   (b)

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

   [1 mark]

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

   (c)

   Find the time at which the particle has speed 24 m s⁻¹.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of mass 0.5 kg enters a viscous liquid with speed 8 m s⁻¹ and moves in a straight line. The only force acting on it in the direction of motion is a resistance of magnitude $2v$ newtons, where $v$ m s⁻¹ is its speed at time $t$ seconds after entering the liquid.

   (a)

   Which equation of motion is correct for the particle?

   [1 mark]

   • A$0.5\frac{dv}{dt}=2v$
   • B$\frac{dv}{dt}=-2v$
   • C$0.5\frac{dv}{dt}=-2v$
   • D$0.5v\frac{dv}{dt}=-2v$

   (b)

   Find the speed of the particle when $t=0.5$.

   [1 mark]

   • A$1.08$ m s⁻¹
   • B$4.85$ m s⁻¹
   • C$0.146$ m s⁻¹
   • D$6$ m s⁻¹

   (c)

   Find the distance travelled by the particle before it comes to rest.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass 2 kg moves along the $x$-axis on a smooth horizontal surface. When $P$ is at the point with coordinate $x$ metres, it is acted on by a force of magnitude $\frac{16}{(x+2)^2}$ newtons directed away from the origin $O$. At $x=0$ the particle has speed 1 m s⁻¹ in the direction of increasing $x$.

   (a)

   Show that $v^2=9-\frac{16}{x+2}$, where $v$ m s⁻¹ is the speed of $P$ at the point with coordinate $x$.

   [3 marks]

   (b)

   Find the speed of $P$ when $x=6$, and show that the speed of $P$ never reaches 3 m s⁻¹, however far it travels.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The Earth is modelled as a fixed sphere of radius $R=6.4\times10^6$ m, with gravitational acceleration $g=9.8$ m s⁻² at its surface. A particle of mass $m$ at a distance $x$ from the centre of the Earth, where $x\ge R$, experiences a gravitational force of magnitude $\frac{mgR^2}{x^2}$ directed towards the centre. A particle is projected vertically upwards from the surface with speed $u$ m s⁻¹. Ignore air resistance and the rotation of the Earth.

   (a)

   Show that, when the particle is at a distance $x$ from the centre of the Earth, $v^2=u^2-2gR+\frac{2gR^2}{x}$, where $v$ m s⁻¹ is its speed. Hence find the greatest height above the surface reached when $u=4000$, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   Find the least value of $u$ for which the particle never returns to the Earth, giving your answer to 3 significant figures, and justify your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions