Variable forces and powerAQA A-Level Further Maths: Subtopic test

1. 1
   A particle of mass 2 kg is moved along a straight line from $x=0$ to $x=4$ by a force of magnitude $F=3x^2+2$ newtons acting in the direction of motion, where $x$ is the distance in metres from the starting point. The particle starts from rest and no other force does work.

   (a)

   Find the work done by the force as the particle moves from $x=0$ to $x=4$.

   [1 mark]

   • A$72$ J
   • B$50$ J
   • C$200$ J
   • D$66$ J

   (b)

   Find the speed of the particle when $x=4$.

   [1 mark]

   • A$6.00\text{ m s}^{-1}$
   • B$36\text{ m s}^{-1}$
   • C$8.49\text{ m s}^{-1}$
   • D$72\text{ m s}^{-1}$

   (c)

   Find the work done by the force as the particle moves from $x=1$ to $x=3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A car of mass 1200 kg travels along a straight horizontal road with its engine working at a constant 30 kW. The resistance to motion is a constant 750 N.

   (a)

   Find the driving force of the engine when the car is moving at $20\text{ m s}^{-1}$.

   [1 mark]

   • A$750$ N
   • B$1500$ N
   • C$1.5$ N
   • D$600\,000$ N

   (b)

   Find the greatest speed the car can maintain on this road.

   [1 mark]

   • A$20\text{ m s}^{-1}$
   • B$25\text{ m s}^{-1}$
   • C$80\text{ m s}^{-1}$
   • D$40\text{ m s}^{-1}$

   (c)

   Find the acceleration of the car when its speed is $20\text{ m s}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A train of mass 200 000 kg moves along a straight horizontal track. The engine works at a constant 400 kW and the resistance to motion is a constant 8000 N.

   (a)

   Find the acceleration of the train when its speed is $25\text{ m s}^{-1}$.

   [3 marks]

   (b)

   The train is travelling at a steady $50\text{ m s}^{-1}$ when the driver reduces the power to 300 kW. Find the deceleration of the train immediately after the reduction, and the speed at which it eventually travels steadily.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass 4 kg moves in a straight line on a smooth horizontal surface. It passes the point $O$ with speed $2\text{ m s}^{-1}$. A force of magnitude $F=(20-4x)$ newtons then acts on $P$ in its direction of motion, where $x$ metres is the distance of $P$ from $O$.

   (a)

   Find the work done by the force as $P$ moves from $x=0$ to $x=3$, and hence find the speed of $P$ when $x=3$.

   [6 marks]

   (b)

   Show that $P$ has its greatest speed at $x=5$, and find this greatest speed.

   [6 marks]

   Total for question 4: 12 marks

End of questions