Conservation of momentumAQA A-Level Further Maths: Subtopic test

1. 1
   Particle $A$ of mass $3$ kg, moving at $4$ m s⁻¹, collides directly with particle $B$ of mass $2$ kg, which is moving in the same direction at $1$ m s⁻¹ on a smooth horizontal surface. The particles coalesce.

   (a)

   Find the total momentum of the two particles before the collision.

   [1 mark]

   • A$14$ kg m s⁻¹
   • B$10$ kg m s⁻¹
   • C$12$ kg m s⁻¹
   • D$5$ kg m s⁻¹

   (b)

   Find the speed of the combined particle after the collision.

   [1 mark]

   • A$2.5$ m s⁻¹
   • B$\frac{14}{3}$ m s⁻¹
   • C$2$ m s⁻¹
   • D$2.8$ m s⁻¹

   (c)

   Find the kinetic energy lost in the collision.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A gun of mass $800$ kg, free to recoil on smooth horizontal ground, fires a shell of mass $8$ kg horizontally. The gun and shell are initially at rest, and the shell leaves the gun with speed $250$ m s⁻¹ relative to the ground.

   (a)

   Find the speed at which the gun begins to recoil.

   [1 mark]

   • A$25$ m s⁻¹
   • B$2.48$ m s⁻¹
   • C$2.5$ m s⁻¹
   • D$0.4$ m s⁻¹

   (b)

   Find the total kinetic energy of the gun and shell just after firing.

   [1 mark]

   • A$250\,000$ J
   • B$252\,500$ J
   • C$2500$ J
   • D$505\,000$ J

   (c)

   Find the speed of the shell relative to the gun.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two smooth spheres $P$ and $Q$, of masses $2m$ and $m$, move towards each other along a straight line on a smooth horizontal table, with speeds $3u$ and $2u$ respectively. After the collision $Q$ moves in the direction of $P$'s original motion, with speed $2u$.

   (a)

   Find the velocity of $P$ after the collision.

   [3 marks]

   (b)

   Find, in terms of $m$ and $u$, the kinetic energy lost in the collision.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $A$ and $B$, of masses $2$ kg and $3$ kg, move on a smooth horizontal plane. Before they collide, $A$ has velocity $(4\mathbf{i}+\mathbf{j})$ m s⁻¹ and $B$ has velocity $(-2\mathbf{i}+3\mathbf{j})$ m s⁻¹, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular horizontal unit vectors. After the collision $A$ has velocity $(\mathbf{i}+\mathbf{j})$ m s⁻¹.

   (a)

   (i) Find the velocity of $B$ after the collision.
   (ii) Find the kinetic energy lost in the collision.

   [6 marks]

   (b)

   After this collision, $B$ collides with a particle $C$ of mass $1$ kg moving with velocity $(4\mathbf{i}-5\mathbf{j})$ m s⁻¹, and they coalesce. Find the speed of the combined particle and the angle its direction makes with $\mathbf{i}$, and find the kinetic energy lost in this second collision.

   [6 marks]

   Total for question 4: 12 marks

End of questions