Mechanics 2 (M2): Work and energyEdexcel International A Level Further Maths: Topic test

1. 1
   A crane lifts a load of mass $250$ kg vertically at constant speed through a height of $12$ m in $20$ s. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the work done by the crane in lifting the load.

   [1 mark]

   • A$3000$ J
   • B$29\,400$ J
   • C$1470$ J
   • D$588\,000$ J

   (b)

   Find the average power developed by the crane during the lift.

   [1 mark]

   • A$588\,000$ W
   • B$2450$ W
   • C$1.47$ W
   • D$1470$ W

   (c)

   The crane now lifts the same load at a constant speed while working at a constant rate of $4.9$ kW. Find this speed.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A smooth bead of mass $0.2$ kg is threaded on a smooth curved wire fixed in a vertical plane. The bead is released from rest at a point $A$, which is $2.5$ m above the lowest point $B$ of the wire. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the speed of the bead at $B$.

   [1 mark]

   • A$7\text{ m s}^{-1}$
   • B$49\text{ m s}^{-1}$
   • C$4.95\text{ m s}^{-1}$
   • D$24.5\text{ m s}^{-1}$

   (b)

   Find the kinetic energy of the bead at $B$.

   [1 mark]

   • A$9.8$ J
   • B$49$ J
   • C$4.9$ J
   • D$1.4$ J

   (c)

   Find the speed of the bead at the point on the wire that is $1$ m above $B$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cyclist and her bicycle, of total mass $90$ kg, ride at a constant speed of $6\text{ m s}^{-1}$ up a straight road inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha=\frac{1}{20}$. The total resistance to motion is a constant $30$ N. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the rate at which the cyclist is working.

   [3 marks]

   (b)

   Later the cyclist freewheels, without pedalling, down the same road, passing a point $A$ with speed $6\text{ m s}^{-1}$. The resistance is still $30$ N. Find her speed after she has travelled a further $80$ m down the road.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A parcel of mass $4$ kg is released from rest at a point $A$ at the top of a smooth ramp. $AB$ is a line of greatest slope of the ramp, with $AB=6$ m, and the ramp is inclined at $30^\circ$ to the horizontal. At $B$ the parcel moves onto rough horizontal ground, where it experiences a constant resistance of $14.7$ N, and it comes to rest at the point $C$. The change of direction at $B$ causes no loss of energy. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the kinetic energy of the parcel at $B$ and the speed of the parcel at $B$.

   [6 marks]

   (b)

   Find the distance $BC$, and find the speed of the parcel when it is $4$ m beyond $B$ on the rough ground.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A van of mass $1800$ kg has an engine working at a constant rate of $36$ kW. On a horizontal road the resistance to motion is a constant $900$ N.

   (a)

   Find the maximum speed of the van on the horizontal road.

   [1 mark]

   • A$0.025\text{ m s}^{-1}$
   • B$80\text{ m s}^{-1}$
   • C$40\text{ m s}^{-1}$
   • D$32\,400\,000\text{ m s}^{-1}$

   (b)

   Find the acceleration of the van when it is travelling at $20\text{ m s}^{-1}$ on the horizontal road.

   [1 mark]

   • A$0.5\text{ m s}^{-2}$
   • B$1\text{ m s}^{-2}$
   • C$1.5\text{ m s}^{-2}$
   • D$-0.5\text{ m s}^{-2}$

   (c)

   Find the work done by the engine in $30$ s while it is working at this constant rate.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A stone of mass $0.4$ kg is thrown vertically upwards from ground level with speed $14\text{ m s}^{-1}$. Air resistance may be ignored. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the greatest height reached by the stone.

   [1 mark]

   • A$20$ m
   • B$14$ m
   • C$7$ m
   • D$10$ m

   (b)

   Find the kinetic energy of the stone when it is $4$ m above the ground.

   [1 mark]

   • A$39.2$ J
   • B$23.5$ J
   • C$15.7$ J
   • D$54.9$ J

   (c)

   Find the speed of the stone when it is $4$ m above the ground.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A sledge of mass $15$ kg is pulled from rest up a line of greatest slope of a rough plane by a constant force of $100$ N acting parallel to the plane. The plane is inclined at an angle $\theta$ to the horizontal, where $\sin\theta=0.2$, and the resistance to motion is a constant $25$ N. After the sledge has moved $6$ m up the plane the rope breaks. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the speed of the sledge at the instant the rope breaks.

   [3 marks]

   (b)

   After the rope breaks the sledge continues up the plane. Find the further distance the sledge travels up the plane before it first comes to rest.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ of mass $0.3$ kg moves on a track in a vertical plane. The section $AB$ of the track is smooth and curved, and $A$ is $1.8$ m above the level of $B$. The section $BC$ is rough and horizontal, with $BC=2.5$ m, and $P$ experiences a constant resistance of $1.47$ N while moving along it. The section $CD$ is a smooth ramp rising from $C$. The particle is released from rest at $A$ and no energy is lost at $B$ or at $C$. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the kinetic energy and the speed of $P$ at $C$.

   [6 marks]

   (b)

   Find the greatest height of $P$ above $BC$ on the ramp $CD$. Hence show that $P$ comes to rest on $BC$, and find its distance from $C$ when it stops.

   [6 marks]

   Total for question 8: 12 marks

End of questions