Variable forcesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle $P$ of mass $2$ kg moves in a straight line. At time $t$ seconds the resultant force on $P$ is $(6t+4)$ N in the direction of motion. $P$ is at rest at the point $O$ when $t=0$.

   (a)

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

   [1 mark]

   • A$16$ m s$^{-2}$
   • B$32$ m s$^{-2}$
   • C$8$ m s$^{-2}$
   • D$3$ m s$^{-2}$

   (b)

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

   [1 mark]

   • A$10$ m s$^{-1}$
   • B$16$ m s$^{-1}$
   • C$20$ m s$^{-1}$
   • D$8$ m s$^{-1}$

   (c)

   Find the distance travelled by $P$ in the first $2$ seconds.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $P$ of mass $0.5$ kg moves along the positive $x$-axis on a smooth horizontal surface, starting from the origin $O$ with speed $6$ m s$^{-1}$. When $P$ is at displacement $x$ metres from $O$, the only horizontal force on $P$ is a resistance of magnitude $2x$ N acting towards $O$.

   (a)

   Which equation of motion applies to $P$, with $v$ its speed at displacement $x$?

   [1 mark]

   • A$\frac{dv}{dx}=-4x$
   • B$v\frac{dv}{dx}=-2x$
   • C$v\frac{dv}{dx}=-x$
   • D$v\frac{dv}{dx}=-4x$

   (b)

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

   [1 mark]

   • A$6$ m
   • B$3$ m
   • C$9$ m
   • D$3\sqrt2$ m

   (c)

   Find the speed of $P$ when it is $2$ m from $O$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass $3$ kg moves in a straight line. At time $t$ seconds, $P$ has speed $v$ m s$^{-1}$ and the only horizontal force acting on $P$ is a resistance of magnitude $6v$ N. When $t=0$ the speed of $P$ is $8$ m s$^{-1}$.

   (a)

   Show that $v=8e^{-2t}$.

   [3 marks]

   (b)

   Find the distance travelled by $P$ while its speed falls from $8$ m s$^{-1}$ to $2$ m s$^{-1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A spacecraft of mass $m$ moves directly away from the centre of the Earth. At distance $x$ from the centre of the Earth, the only force on the spacecraft is the gravitational attraction $\frac{mgR^2}{x^2}$ towards the centre, where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity at the Earth's surface. When $x=2R$ the spacecraft has speed $u$, directed away from the Earth.

   (a)

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

   [6 marks]

   (b)

   Find the least value of $u$, in terms of $g$ and $R$, for which the spacecraft never returns to the Earth. Hence calculate this value in km s$^{-1}$, given that $R=6.4\times10^6$ m and $g=9.8$ m s$^{-2}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions