Mechanics 2 (M2): CollisionsEdexcel International A Level Further Maths: Topic test

1. 1
   A particle $P$ of mass $2$ kg is moving with velocity $(3\mathbf{i}+4\mathbf{j})$ m s$^{-1}$ when it receives an impulse of $(-10\mathbf{i}+6\mathbf{j})$ N s.

   (a)

   Find the velocity of $P$ immediately after it receives the impulse, in m s$^{-1}$.

   [1 mark]

   • A$-7\mathbf{i}+10\mathbf{j}$
   • B$-2\mathbf{i}+7\mathbf{j}$
   • C$8\mathbf{i}+\mathbf{j}$
   • D$-5\mathbf{i}+3\mathbf{j}$

   (b)

   Find the speed of $P$ immediately after it receives the impulse, in m s$^{-1}$ to 3 significant figures.

   [1 mark]

   • A$11.7$
   • B$5.83$
   • C$7.28$
   • D$12.2$

   (c)

   Find the angle through which the direction of motion of $P$ is turned by the impulse, giving your answer in degrees to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two smooth spheres $M$ and $N$ of equal radii, of masses $4$ kg and $1$ kg, move towards each other along the same straight line on a smooth horizontal table. Sphere $M$ has speed $5$ m s$^{-1}$ and sphere $N$ has speed $2$ m s$^{-1}$. They collide directly. Immediately after the collision, the speed of $M$ is $3$ m s$^{-1}$ and its direction of motion is unchanged.

   (a)

   Find the speed of $N$ immediately after the collision, in m s$^{-1}$.

   [1 mark]

   • A$6$
   • B$10$
   • C$8$
   • D$3.6$

   (b)

   Find the coefficient of restitution between $M$ and $N$.

   [1 mark]

   • A$\frac73$
   • B$1$
   • C$\frac67$
   • D$\frac37$

   (c)

   Find the magnitude of the impulse exerted by $M$ on $N$ in the collision.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two smooth spheres $S$ and $T$ of equal radii, of masses $1$ kg and $3$ kg, are on a smooth horizontal floor. Sphere $T$ is at rest and nearer to a fixed smooth vertical wall, and the line of centres of $S$ and $T$ is perpendicular to the wall. Sphere $S$ is projected directly towards $T$ with speed $8$ m s$^{-1}$ and collides with it. The coefficient of restitution between $S$ and $T$ is $\frac12$.

   (a)

   Find the velocity of each sphere immediately after the collision, stating the direction of motion of each.

   [3 marks]

   (b)

   The coefficient of restitution between $T$ and the wall is $\frac12$. Find the speed of $T$ immediately after it hits the wall, and show that there is a second collision between $S$ and $T$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two particles $U$ and $V$, of masses $0.4$ kg and $0.2$ kg, lie at rest on a smooth horizontal surface in a straight line perpendicular to a fixed smooth vertical wall, with $V$ between $U$ and the wall. Particle $U$ is given an impulse of magnitude $4.8$ N s directly towards $V$ and then collides directly with $V$. The coefficient of restitution between $U$ and $V$ is $\frac12$ and the coefficient of restitution between $V$ and the wall is $\frac14$.

   (a)

   (i) Find the speed of $U$ immediately before it collides with $V$.
   (ii) Find the speed of each particle immediately after this collision.

   [6 marks]

   (b)

   (i) Find the speed of $V$ immediately after it hits the wall.
   (ii) Show that $U$ and $V$ collide again, and find the speed of each particle immediately after this second collision.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Two particles $C$ and $D$, of masses $2$ kg and $3$ kg, are moving on a smooth horizontal plane. Just before they collide, the velocity of $C$ is $(4\mathbf{i}-\mathbf{j})$ m s$^{-1}$ and the velocity of $D$ is $(-\mathbf{i}+3\mathbf{j})$ m s$^{-1}$. The particles coalesce to form a single particle.

   (a)

   Find the velocity of the combined particle, in m s$^{-1}$.

   [1 mark]

   • A$1.5\mathbf{i}+\mathbf{j}$
   • B$5\mathbf{i}+7\mathbf{j}$
   • C$2.2\mathbf{i}-2.2\mathbf{j}$
   • D$\mathbf{i}+1.4\mathbf{j}$

   (b)

   Find the speed of the combined particle, in m s$^{-1}$ to 3 significant figures.

   [1 mark]

   • A$1.72$
   • B$8.60$
   • C$1.80$
   • D$2.96$

   (c)

   Find the magnitude of the impulse exerted by $C$ on $D$ in the collision, in N s to 3 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sphere $H$ of mass $1$ kg moving with speed $8$ m s$^{-1}$ collides directly with a stationary sphere $K$ of mass $3$ kg on a smooth horizontal surface. The spheres have equal radii. The coefficient of restitution between $H$ and $K$ is $e$.

   (a)

   Given that $e=\frac12$, find the speed of $K$ immediately after the collision, in m s$^{-1}$.

   [1 mark]

   • A$2$
   • B$3$
   • C$4$
   • D$1$

   (b)

   Given that $e=\frac12$, find the kinetic energy lost in the collision, in J.

   [1 mark]

   • A$14$
   • B$32$
   • C$18$
   • D$13.5$

   (c)

   Find the set of values of $e$ for which the direction of motion of $H$ is reversed by the collision.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Three smooth spheres $X$, $Y$ and $Z$ of equal radii, of masses $m$, $2m$ and $4m$ respectively, lie in a straight line on a smooth horizontal table with $Y$ between $X$ and $Z$. Spheres $Y$ and $Z$ are at rest. Sphere $X$ is projected directly towards $Y$ with speed $12$ m s$^{-1}$. The coefficient of restitution between any two of the spheres is $\frac12$.

   (a)

   Find the speed of $X$ and the speed of $Y$ immediately after the first collision.

   [3 marks]

   (b)

   Find the speeds of $Y$ and $Z$ immediately after $Y$ collides with $Z$, and state, with a reason, whether any further collisions occur.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two smooth spheres $G$ and $H$ of equal radii, of masses $3m$ and $m$, are moving towards each other along the same straight line on a smooth horizontal table. Sphere $G$ has speed $4u$ and sphere $H$ has speed $2u$. The spheres collide directly. The coefficient of restitution between $G$ and $H$ is $e$.

   (a)

   Find, in terms of $u$ and $e$, the velocity of each sphere immediately after the collision, taking the original direction of motion of $G$ as positive.

   [6 marks]

   (b)

   The total kinetic energy lost in the collision is $12mu^2$.
   (i) Show that $e=\frac13$.
   (ii) After the collision, $H$ strikes a smooth fixed wall that is perpendicular to the line of motion and ahead of $H$. The coefficient of restitution between $H$ and the wall is $\frac12$. Find the speed of $H$ immediately after it hits the wall, and show that $G$ and $H$ collide again.

   [6 marks]

   Total for question 8: 12 marks

End of questions